7. Vectors in 2- dimensions

# 7. Vectors in 2- dimensions ## Scalar and vector quantities, Vector notation and representation. ### Introduction Welcome to the exciting world of vectors! In our daily lives and in many scientific fields, we often deal with quantities that have both size and direction. For example, when a plane flies, its movement isn't just about how fast it's going (speed), but also *where* it's going (direction). This lesson will introduce you to two fundamental types of physical quantities: scalars and vectors, and teach you how to represent and understand them. ### Key Concepts #### Scalar Quantities Scalar quantities are physical quantities that are completely described by their **magnitude** (size or amount) alone. They do not have a direction. * **Definition:** A quantity that has only magnitude. * **Examples:** * **Mass:** The amount of matter in an object (e.g., 5 kg). * **Time:** Duration of an event (e.g., 10 seconds). * **Temperature:** Degree of hotness or coldness (e.g., 25 degrees Celsius). * **Speed:** How fast an object is moving, without considering direction (e.g., 60 km/h). * **Distance:** The total path covered by an object (e.g., 100 meters). * **Volume:** The amount of space an object occupies (e.g., 500 mL). #### Vector Quantities Vector quantities are physical quantities that require both **magnitude** and **direction** for their complete description. * **Definition:** A quantity that has both magnitude and direction. * **Examples:** * **Displacement:** The change in position of an object from a starting point, including direction (e.g., 5 km North). * **Velocity:** Speed in a given direction (e.g., 60 km/h East). * **Acceleration:** The rate of change of velocity (e.g., 9.8 m/s² downwards). * **Force:** A push or pull with a certain strength and direction (e.g., 100 Newtons upwards). * **Momentum:** The product of mass and velocity. #### Vector Notation To distinguish vectors from scalars, we use special notation. * **Representing Vectors:** * **Bold lowercase letters:** Such as \( \mathbf{a} \), \( \mathbf{b} \), \( \mathbf{v} \). * **Lowercase letters with an arrow above:** Such as \( \vec{a} \), \( \vec{b} \), \( \vec{v} \). * **Capital letters for start and end points with an arrow above:** If a vector starts at point A and ends at point B, it is denoted as \( \vec{AB} \). * **Magnitude of a Vector:** * The magnitude (or length) of a vector \( \mathbf{a} \) is denoted as \( |\mathbf{a}| \) or \( |\vec{a}| \). * Sometimes, the magnitude is simply written as 'a' (without bolding or an arrow) when the context is clear. * For a vector \( \vec{AB} \), its magnitude is the length of the line segment AB. #### Vector Representation ##### Geometrical Representation Vectors are commonly represented geometrically as directed line segments (arrows). * **Arrow:** The arrow indicates the **direction** of the vector. * **Length:** The length of the arrow represents the **magnitude** of the vector. * **Initial Point (Tail):** The starting point of the vector. * **Terminal Point (Head):** The ending point of the vector, where the arrow is. ##### Component Form (Cartesian Coordinates) In a 2-dimensional Cartesian coordinate system, a vector can be represented by its components along the x and y axes. * **From the Origin:** A vector from the origin O(0,0) to a point P(x,y) is called a **position vector** and is written as: \[ \vec{OP} = \begin{pmatrix} x \\ y \end{pmatrix} \quad \text{or} \quad \vec{OP} = x\mathbf{i} + y\mathbf{j} \] where \( \mathbf{i} \) and \( \mathbf{j} \) are unit vectors along the x and y axes, respectively. * **Between Two Points:** If a vector starts at point \( A(x_1, y_1) \) and ends at point \( B(x_2, y_2) \), its component form is given by: \[ \vec{AB} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix} \] * **Magnitude in Component Form:** For a vector \( \mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix} \), its magnitude is calculated using the Pythagorean theorem: \[ |\mathbf{v}| = \sqrt{x^2 + y^2} \] #### Equal Vectors Two vectors are considered equal if they have the same magnitude and the same direction, regardless of their starting position. They are parallel and of the same length. #### Negative Vectors A negative vector of a given vector has the same magnitude but points in the opposite direction. If \( \vec{a} \) is a vector, then \( -\vec{a} \) is its negative vector. ### Vocabulary * **Scalar:** A quantity with only magnitude. * **Vector:** A quantity with both magnitude and direction. * **Magnitude:** The size or amount of a quantity. * **Direction:** The orientation or path of a vector. * **Initial Point (Tail):** The starting point of a vector. * **Terminal Point (Head):** The ending point of a vector, indicated by the arrow. * **Component Form:** Representation of a vector using its x and y components in a coordinate system. * **Position Vector:** A vector that starts from the origin (0,0). ### Examples 1. **Identifying Scalars and Vectors:** * **Question:** Classify the following as scalar or vector quantities: * a) 20 meters * b) 15 km/h South * c) 5 kilograms * d) A force of 50 N pulling a door open * **Solution:** * a) 20 meters (Distance) - **Scalar** * b) 15 km/h South (Velocity) - **Vector** * c) 5 kilograms (Mass) - **Scalar** * d) A force of 50 N pulling a door open (Force) - **Vector** 2. **Representing a Vector Geometrically:** * **Question:** Draw a vector representing a displacement of 4 units to the East. * **Solution:** Draw an arrow pointing horizontally to the right (East) with a length representing 4 units. Label the initial point A and the terminal point B, so it represents \( \vec{AB} \). 3. **Finding Component Form and Magnitude:** * **Question:** Given points \( P(1, 2) \) and \( Q(4, 6) \), find the vector \( \vec{PQ} \) in component form and its magnitude. * **Solution:** * To find the component form \( \vec{PQ} \): \[ \vec{PQ} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix} = \begin{pmatrix} 4 - 1 \\ 6 - 2 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \] * To find the magnitude \( |\vec{PQ}| \): \[ |\vec{PQ}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ units} \] ### Exercises 1. **Classification:** Classify each of the following quantities as either scalar or vector: * a) The speed of a car is 80 km/h. * b) A child's weight is 300 N downwards. * c) The room temperature is 22°C. * d) A plane travels 500 km North-West. * e) It took 2 hours to complete the journey. 2. **Geometrical Representation:** Draw a vector representing each of the following: * a) A displacement of 3 cm at an angle of 30° from the positive x-axis. * b) A force of 20 N vertically upwards. 3. **Component Form and Magnitude:** Given the points \( A(2, 1) \) and \( B(7, 13) \): * a) Write down the column vector \( \vec{AB} \). * b) Calculate the magnitude of vector \( \vec{AB} \). 4. **Magnitude of a Vector:** If vector \( \mathbf{v} = \begin{pmatrix} -6 \\ 8 \end{pmatrix} \), find \( |\mathbf{v}| \). ### Assignment 1. **Scalar and Vector Identification:** List three scalar quantities and three vector quantities that were not explicitly mentioned in the lesson notes. 2. **Vector Components and Magnitude:** A hiker walks from point \( X(5, 2) \) to point \( Y(-3, 8) \). * a) Express the displacement vector \( \vec{XY} \) in column form. * b) Calculate the total distance (magnitude of displacement) the hiker is from the starting point \( X \). 3. **Equal and Negative Vectors:** * a) Given vector \( \mathbf{p} = \begin{pmatrix} 2 \\ -5 \end{pmatrix} \), write down a vector \( \mathbf{q} \) that is equal to \( \mathbf{p} \). * b) Given vector \( \mathbf{r} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} \), write down the negative vector \( -\mathbf{r} \). ### Summary In this lesson, we explored the fundamental difference between scalar and vector quantities. Scalar quantities, like mass and time, are described by magnitude only. Vector quantities, such as displacement and force, require both magnitude and direction for their complete description. We learned how to denote vectors using bold letters or arrows, and how to represent them geometrically with directed line segments. We also saw how to express vectors in component form \( \begin{pmatrix} x \\ y \end{pmatrix} \) using Cartesian coordinates and how to calculate their magnitude using the formula \( \sqrt{x^2 + y^2} \). Understanding these basics is crucial for further studies in physics and higher mathematics. ## Position vectors, free vectors and localized vectors. ### Introduction Welcome to this lesson on vectors! In our previous studies, we learned that vectors are quantities that have both magnitude (size) and direction, unlike scalars which only have magnitude. Today, we will dive deeper into the world of vectors by exploring different types: position vectors, free vectors, and localized vectors. Understanding these distinctions is crucial for applying vectors correctly in mathematics and other subjects like physics and engineering. ### Key Concepts #### What is a Vector? (A Quick Recap) A vector is a mathematical object that has both a magnitude (length or size) and a direction. We often represent a vector with an arrow, where the length of the arrow represents its magnitude and the way it points indicates its direction. Contrast this with a **scalar**, which is a quantity that only has magnitude (e.g., mass, temperature, time). A vector, on the other hand, tells you 'how much' and 'in what direction'. #### Position Vectors ##### Definition A position vector is a special type of vector that specifies the position of a point in space relative to a fixed origin. In a 2-dimensional coordinate system, the origin is usually the point \( (0,0) \). If point P has coordinates \( (x, y) \), its position vector starts from the origin O \( (0,0) \) and ends at point P. It is commonly denoted as \( \vec{OP} \) or simply \( \mathbf{p} \). ##### Notation and Representation The position vector \( \vec{OP} \) for a point \( P(x, y) \) can be written as a column vector: \[ \vec{OP} = \begin{pmatrix} x \\ y \end{pmatrix} \] ##### Magnitude of a Position Vector The magnitude (or length) of the position vector \( \vec{OP} = \begin{pmatrix} x \\ y \end{pmatrix} \) is found using the Pythagorean theorem, as it represents the distance from the origin to point P: \[ |\vec{OP}| = \sqrt{x^2 + y^2} \] #### Free Vectors ##### Definition A free vector is a vector that has a specific magnitude and direction but does **not** have a fixed starting point or point of application. It represents a translation or displacement. ##### Characteristics * Free vectors can be moved freely in space as long as their magnitude and direction remain unchanged. * Two free vectors are considered equal if they have the same magnitude and the same direction, regardless of where they start or end. * For example, if you move 5 km North, that displacement is a free vector. It doesn't matter if you started from your home or your school; the displacement vector itself is the same. ##### Representing a Free Vector between Two Points If you have two points, A and B, with position vectors \( \vec{OA} = \mathbf{a} \) and \( \vec{OB} = \mathbf{b} \) respectively, the free vector representing the displacement from A to B is given by: \[ \vec{AB} = \vec{OB} - \vec{OA} = \mathbf{b} - \mathbf{a} \] #### Localized Vectors ##### Definition A localized vector is a vector that has a specific magnitude, direction, **and** a fixed point of application. Its effect is tied to where it is applied. ##### Characteristics * Localized vectors cannot be moved without changing their effect or the problem they describe. * They are often used in physics to represent quantities like force, where the point of application significantly impacts the outcome (e.g., pushing a door near the hinges versus near the handle). * The velocity of a specific particle at a precise moment in time is also a localized vector. #### Relationship between Position, Free, and Localized Vectors * A position vector is essentially a special type of localized vector, always localized at the origin. * A free vector can be represented by a position vector if we imagine its tail (starting point) is at the origin. * The key differentiator is the importance of the starting point or point of application. If it matters, it's either a localized or position vector. If it doesn't, it's a free vector. ### Vocabulary * **Vector:** A quantity possessing both magnitude and direction. * **Scalar:** A quantity possessing only magnitude. * **Magnitude:** The length or size of a vector. * **Direction:** The orientation of a vector in space. * **Origin:** The fixed reference point \( (0,0) \) in a coordinate system. * **Position Vector:** A vector that describes the position of a point relative to the origin. * **Free Vector:** A vector defined by its magnitude and direction only, without a fixed starting point. It can be translated anywhere in space. * **Localized Vector:** A vector defined by its magnitude, direction, and a specific point of application. Its effect depends on where it is applied. ### Examples #### Example 1: Position Vectors and their Magnitudes Given points A(3, 2), B(-1, 4), and C(0, -5). 1. Find the position vectors \( \vec{OA} \), \( \vec{OB} \), and \( \vec{OC} \). 2. Find the magnitudes of these position vectors. **Solution:** 1. \[ \vec{OA} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} \] \[ \vec{OB} = \begin{pmatrix} -1 \\ 4 \end{pmatrix} \] \[ \vec{OC} = \begin{pmatrix} 0 \\ -5 \end{pmatrix} \] 2. \[ |\vec{OA}| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \] \[ |\vec{OB}| = \sqrt{(-1)^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \] \[ |\vec{OC}| = \sqrt{0^2 + (-5)^2} = \sqrt{0 + 25} = \sqrt{25} = 5 \] #### Example 2: Free Vectors Points P are at (1, 2) and Q are at (4, 6). 1. Find the free vector \( \vec{PQ} \). 2. If R is at (0, 0) and S is at (3, 4), is \( \vec{PQ} \) equal to \( \vec{RS} \)? Explain. **Solution:** 1. To find \( \vec{PQ} \), we use the position vectors of Q and P: \[ \vec{PQ} = \vec{OQ} - \vec{OP} = \begin{pmatrix} 4 \\ 6 \end{pmatrix} - \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 4-1 \\ 6-2 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \] 2. To find \( \vec{RS} \): \[ \vec{RS} = \vec{OS} - \vec{OR} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} - \begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \] Yes, \( \vec{PQ} \) is equal to \( \vec{RS} \) because they both have the same component form \( \begin{pmatrix} 3 \\ 4 \end{pmatrix} \). This means they represent the same magnitude and direction, hence they are the same free vector. #### Example 3: Localized vs. Free Describe a scenario where a vector would be considered localized and another where it would be considered free. **Solution:** * **Localized Vector Scenario:** A mechanic uses a wrench to tighten a bolt. The force applied to the end of the wrench is a localized vector. The *effect* of that force (the torque, or turning force) depends critically on *where* on the wrench the force is applied (closer to the bolt or further from it) and *in which direction*. * **Free Vector Scenario:** A bird flies 10 km east. This displacement is a free vector. It represents a journey of a specific distance in a specific direction. It doesn't matter where the bird started its flight (e.g., from its nest or a tree); the 10 km east displacement is the same vector. ### Exercises 1. Point A is at \( (5, -3) \). Write its position vector \( \vec{OA} \) and find its magnitude. 2. Points X(2, 7) and Y(8, 1). Find the free vector \( \vec{XY} \). 3. Given \( \mathbf{u} = \begin{pmatrix} -2 \\ 5 \end{pmatrix} \) and \( \mathbf{v} = \begin{pmatrix} -2 \\ 5 \end{pmatrix} \). Are \( \mathbf{u} \) and \( \mathbf{v} \) necessarily the same localized vector? Are they necessarily the same free vector? Explain your reasoning. 4. A force of 10 N is applied at an angle of 30 degrees to the horizontal *at the top-left corner of a box*. Is this best described as a position vector, a free vector, or a localized vector? Why? ### Assignment 1. If P is at \( (-4, 6) \) and Q is at \( (2, -1) \), find: * a) The position vector \( \vec{OP} \). * b) The position vector \( \vec{OQ} \). * c) The free vector \( \vec{PQ} \). * d) The magnitude of \( \vec{PQ} \). 2. Points S(7, -2), T(1, 3), U(-5, 0), V(-11, 5). * a) Find the vector \( \vec{ST} \). * b) Find the vector \( \vec{VU} \). * c) Show that \( \vec{ST} \) is parallel and equal in magnitude to \( \vec{VU} \). What does this imply about them as free vectors? 3. Classify each of the following quantities as a localized vector, a free vector, or a scalar, providing a brief reason for your choice: * a) The velocity of a bullet fired from a gun at the moment it leaves the barrel. * b) The amount of sugar in a cup of tea. * c) The force exerted by a magnet on a metal object. * d) The displacement from your school to your home. * e) The temperature of the classroom. ### Summary In this lesson, we have explored the three main types of vectors: position vectors, free vectors, and localized vectors. We learned that: * A **position vector** starts from the origin and points to a specific point, defining its location. * A **free vector** is defined only by its magnitude and direction; its starting point is not fixed and it can represent a pure displacement. * A **localized vector** has a specific magnitude, direction, AND a fixed point of application, meaning its effect depends on where it is applied. Understanding these distinctions is crucial for solving vector problems and applying them in real-world contexts, especially in physics and engineering where the point of application of a force or velocity can be very important. ## Operations with vectors, Equal vectors, Addition and subtraction of vectors ### Introduction Welcome to this lesson on vectors! In our previous lessons, we learned about what vectors are and how they represent quantities with both magnitude (size) and direction. In this lesson, we will dive into the fundamental operations we can perform with vectors, such as determining if two vectors are equal, and how to add and subtract them. Understanding these operations is crucial for solving problems in physics, engineering, and many other fields. ### Key Concepts #### Representation of Vectors Before we begin with operations, let's quickly recall how vectors are represented: * **Column Vector:** A vector can be represented as a column of numbers, often like \( \begin{pmatrix} x \\ y \end{pmatrix} \), where \(x\) is the horizontal component and \(y\) is the vertical component. * **Position Vector:** A position vector describes the position of a point relative to the origin. For a point \(P(x, y)\), its position vector is \(\vec{OP} = \begin{pmatrix} x \\ y \end{pmatrix}\). * **Vector between two points:** If you have two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the vector \(\vec{AB}\) is given by: \[ \vec{AB} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix} \] #### Equal Vectors Two vectors are considered **equal** if and only if they have: 1. The same **magnitude** (length). 2. The same **direction**. If two vectors \(\mathbf{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}\) are equal, then their corresponding components must be equal: \[ x_1 = x_2 \quad \text{and} \quad y_1 = y_2 \] #### Zero Vector * The **zero vector**, denoted by \(\mathbf{0}\) or \(\vec{0}\), is a vector with zero magnitude and no specific direction. In column form, it is \(\begin{pmatrix} 0 \\ 0 \end{pmatrix}\). #### Negative of a Vector * The **negative of a vector** \(\mathbf{a}\) is a vector that has the same magnitude as \(\mathbf{a}\) but points in the opposite direction. It is denoted by \(-\mathbf{a}\). * If \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\), then \(-\mathbf{a} = \begin{pmatrix} -x \\ -y \end{pmatrix}\). #### Addition of Vectors ##### Geometrical Method There are two main geometrical methods for adding vectors: 1. **Triangle Law of Vector Addition:** If two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are represented by two sides of a triangle taken in the same order, then their sum (resultant vector) is represented by the third side of the triangle taken in the opposite order. * Place the tail of vector \(\mathbf{b}\) at the head of vector \(\mathbf{a}\). * The resultant vector \(\mathbf{a} + \mathbf{b}\) is drawn from the tail of \(\mathbf{a}\) to the head of \(\mathbf{b}\). 2. **Parallelogram Law of Vector Addition:** If two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are represented by the two adjacent sides of a parallelogram, both originating from the same point, then their sum (resultant vector) is represented by the diagonal of the parallelogram that starts from the same common point. ##### Column Vector Method When vectors are given in column form, addition is straightforward: you simply add their corresponding components. If \(\mathbf{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}\), then: \[ \mathbf{a} + \mathbf{b} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix} \] #### Subtraction of Vectors ##### Geometrical Method Subtracting a vector \(\mathbf{b}\) from a vector \(\mathbf{a}\) is equivalent to adding the negative of \(\mathbf{b}\) to \(\mathbf{a}\). \(\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b})\). * First, find the negative of vector \(\mathbf{b}\), which is \(-\mathbf{b}\) (same magnitude, opposite direction). * Then, apply the triangle law or parallelogram law to add \(\mathbf{a}\) and \(-\mathbf{b}\). ##### Column Vector Method Similar to addition, when vectors are in column form, you subtract their corresponding components. If \(\mathbf{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}\), then: \[ \mathbf{a} - \mathbf{b} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} - \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 - x_2 \\ y_1 - y_2 \end{pmatrix} \] ### Vocabulary * **Vector:** A quantity having both magnitude and direction. * **Scalar:** A quantity having only magnitude (e.g., mass, time, temperature). * **Magnitude:** The length or size of a vector. * **Direction:** The orientation of a vector in space. * **Equal Vectors:** Vectors with the same magnitude and the same direction. * **Zero Vector:** A vector with zero magnitude and no specific direction, \(\begin{pmatrix} 0 \\ 0 \end{pmatrix}\). * **Negative of a Vector:** A vector with the same magnitude but opposite direction to the original vector. * **Resultant Vector:** The single vector that represents the sum of two or more vectors. ### Examples #### Example 1: Equal Vectors Given vectors \(\mathbf{p} = \begin{pmatrix} 3 \\ k \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} m \\ -5 \end{pmatrix}\). If \(\mathbf{p} = \mathbf{q}\), find the values of \(k\) and \(m\). **Solution:** For \(\mathbf{p}\) and \(\mathbf{q}\) to be equal, their corresponding components must be equal. \(3 = m \Rightarrow m = 3\) \(k = -5\) Thus, \(k = -5\) and \(m = 3\). #### Example 2: Vector Addition Given \(\mathbf{a} = \begin{pmatrix} 4 \\ -2 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}\). Find \(\mathbf{a} + \mathbf{b}\). **Solution:** \[ \mathbf{a} + \mathbf{b} = \begin{pmatrix} 4 \\ -2 \end{pmatrix} + \begin{pmatrix} -1 \\ 5 \end{pmatrix} = \begin{pmatrix} 4 + (-1) \\ -2 + 5 \end{pmatrix} = \begin{pmatrix} 3 \\ 3 \end{pmatrix} \] #### Example 3: Vector Subtraction Given \(\mathbf{c} = \begin{pmatrix} 7 \\ 3 \end{pmatrix}\) and \(\mathbf{d} = \begin{pmatrix} 2 \\ -4 \end{pmatrix}\). Find \(\mathbf{c} - \mathbf{d}\). **Solution:** \[ \mathbf{c} - \mathbf{d} = \begin{pmatrix} 7 \\ 3 \end{pmatrix} - \begin{pmatrix} 2 \\ -4 \end{pmatrix} = \begin{pmatrix} 7 - 2 \\ 3 - (-4) \end{pmatrix} = \begin{pmatrix} 5 \\ 3 + 4 \end{pmatrix} = \begin{pmatrix} 5 \\ 7 \end{pmatrix} \] #### Example 4: Combined Operations Given \(\mathbf{u} = \begin{pmatrix} 1 \\ 6 \end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix} -3 \\ 2 \end{pmatrix}\) and \(\mathbf{w} = \begin{pmatrix} 0 \\ -4 \end{pmatrix}\). Find \(\mathbf{u} - \mathbf{v} + \mathbf{w}\). **Solution:** \[ \mathbf{u} - \mathbf{v} + \mathbf{w} = \begin{pmatrix} 1 \\ 6 \end{pmatrix} - \begin{pmatrix} -3 \\ 2 \end{pmatrix} + \begin{pmatrix} 0 \\ -4 \end{pmatrix} \] \[ = \begin{pmatrix} 1 - (-3) + 0 \\ 6 - 2 + (-4) \end{pmatrix} \] \[ = \begin{pmatrix} 1 + 3 + 0 \\ 6 - 2 - 4 \end{pmatrix} \] \[ = \begin{pmatrix} 4 \\ 0 \end{pmatrix} \] ### Exercises 1. Given \(\mathbf{a} = \begin{pmatrix} x \\ 7 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -2 \\ y \end{pmatrix}\). If \(\mathbf{a} = \mathbf{b}\), find the values of \(x\) and \(y\). 2. Let \(\mathbf{p} = \begin{pmatrix} 5 \\ -3 \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} 1 \\ 8 \end{pmatrix}\). Calculate: a) \(\mathbf{p} + \mathbf{q}\) b) \(\mathbf{p} - \mathbf{q}\) c) \(\mathbf{q} - \mathbf{p}\) 3. Point A is at \((2, 5)\) and point B is at \((8, -1)\). Find the vector \(\vec{AB}\). 4. If \(\mathbf{m} = \begin{pmatrix} -4 \\ 0 \end{pmatrix}\) and \(\mathbf{n} = \begin{pmatrix} 6 \\ -7 \end{pmatrix}\), find \(\mathbf{m} + \mathbf{n}\). 5. Given vectors \(\mathbf{x} = \begin{pmatrix} 10 \\ 1 \end{pmatrix}\), \(\mathbf{y} = \begin{pmatrix} 3 \\ -5 \end{pmatrix}\) and \(\mathbf{z} = \begin{pmatrix} -6 \\ 4 \end{pmatrix}\). Find \(\mathbf{x} + \mathbf{y} - \mathbf{z}\). ### Assignment 1. Points P, Q, R, and S have coordinates \(P(1, 2)\), \(Q(4, 5)\), \(R(6, 3)\) and \(S(x, y)\) respectively. If \(\vec{PQ} = \vec{RS}\), find the coordinates of point S. 2. Let \(\mathbf{f} = \begin{pmatrix} -9 \\ 12 \end{pmatrix}\) and \(\mathbf{g} = \begin{pmatrix} 7 \\ -3 \end{pmatrix}\). Calculate: a) \(\mathbf{f} + \mathbf{g}\) b) \(\mathbf{f} - \mathbf{g}\) c) \(\mathbf{g} - \mathbf{f}\) 3. Consider vectors \(\mathbf{u} = \begin{pmatrix} 2 \\ -8 \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} -2 \\ 8 \end{pmatrix}\). Are \(\mathbf{u}\) and \(-\mathbf{v}\) equal? Justify your answer. 4. A boat travels with a velocity vector \(\mathbf{v_1} = \begin{pmatrix} 10 \\ 0 \end{pmatrix}\) km/h (east). The river current has a velocity vector \(\mathbf{v_2} = \begin{pmatrix} 0 \\ 2 \end{pmatrix}\) km/h (north). What is the resultant velocity vector of the boat? 5. If \(\begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}\), find the values of \(a\) and \(b\). ### Summary In this lesson, we explored the fundamental operations with vectors in 2-dimensions. We learned that two vectors are **equal** if they have identical components. Vector **addition** involves adding corresponding components (or using the triangle/parallelogram law geometrically), while vector **subtraction** involves subtracting corresponding components (or adding the negative of the vector). These basic operations are the building blocks for more advanced vector concepts and their applications in various fields. ## Multiplication of a vector by a scalar and dot product of two vectors ### Introduction Welcome to this lesson on vectors! Today, we will explore two fundamental operations involving vectors: multiplying a vector by a scalar (a simple number) and finding the dot product of two vectors. These operations are crucial for understanding how vectors behave and for solving various problems in physics, engineering, and other fields. ### Key Concepts #### Multiplication of a Vector by a Scalar ##### Definition When we multiply a vector \( \mathbf{a} \) by a scalar \( k \) (a real number), we get a new vector, denoted as \( k\mathbf{a} \). If \( \mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} \), then \( k\mathbf{a} = k\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix} \). ##### Effect on Magnitude and Direction * **Magnitude:** The magnitude of \( k\mathbf{a} \) is \( |k| \) times the magnitude of \( \mathbf{a} \). \[ |k\mathbf{a}| = |k| |\mathbf{a}| \] * **Direction:** * If \( k > 0 \), then \( k\mathbf{a} \) has the same direction as \( \mathbf{a} \). * If \( k < 0 \), then \( k\mathbf{a} \) has the opposite direction to \( \mathbf{a} \). * If \( k = 0 \), then \( k\mathbf{a} = \mathbf{0} \) (the zero vector). ##### Parallel Vectors Two non-zero vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel if and only if \( \mathbf{b} = k\mathbf{a} \) for some scalar \( k \). If \( k > 0 \), they are in the same direction. If \( k < 0 \), they are in opposite directions. #### Dot Product of Two Vectors (Scalar Product) ##### Definition (Algebraic) The dot product of two vectors \( \mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} \) is a scalar quantity defined as: \[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \] ##### Definition (Geometric) The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) can also be defined using their magnitudes and the angle \( \theta \) between them: \[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \] where \( 0^\circ \le \theta \le 180^\circ \). ##### Properties of the Dot Product * **Commutative Property:** \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \) * **Distributive Property:** \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} \) * **Scalar Multiplication:** \( (k\mathbf{a}) \cdot \mathbf{b} = k(\mathbf{a} \cdot \mathbf{b}) = \mathbf{a} \cdot (k\mathbf{b}) \) * **Dot Product with Itself:** \( \mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2 \) * **Dot Product of Perpendicular Vectors:** If \( \mathbf{a} \) and \( \mathbf{b} \) are non-zero perpendicular vectors, then \( \mathbf{a} \cdot \mathbf{b} = 0 \) (because \( \cos 90^\circ = 0 \)). * **Angle Between Two Vectors:** We can find the angle \( \theta \) between two non-zero vectors using: \[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \] ### Vocabulary * **Scalar:** A quantity that has only magnitude (e.g., a number, temperature, mass). * **Vector:** A quantity that has both magnitude and direction. * **Scalar Multiplication:** The operation of multiplying a vector by a scalar. * **Dot Product (Scalar Product):** An operation that takes two vectors and returns a single scalar quantity. * **Magnitude:** The length or size of a vector. * **Parallel Vectors:** Vectors that have the same or opposite direction. * **Perpendicular Vectors:** Vectors that intersect at a 90-degree angle. ### Examples #### Example 1: Scalar Multiplication Given vector \( \mathbf{v} = \begin{pmatrix} 3 \\ -2 \end{pmatrix} \). Find \( 2\mathbf{v} \) and \( -3\mathbf{v} \). * **Solution:** \[ 2\mathbf{v} = 2\begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 2 \times 3 \\ 2 \times (-2) \end{pmatrix} = \begin{pmatrix} 6 \\ -4 \end{pmatrix} \] \[ -3\mathbf{v} = -3\begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} -3 \times 3 \\ -3 \times (-2) \end{pmatrix} = \begin{pmatrix} -9 \\ 6 \end{pmatrix} \] #### Example 2: Checking for Parallel Vectors Are vectors \( \mathbf{p} = \begin{pmatrix} 4 \\ 6 \end{pmatrix} \) and \( \mathbf{q} = \begin{pmatrix} -2 \\ -3 \end{pmatrix} \) parallel? * **Solution:** We need to check if \( \mathbf{p} = k\mathbf{q} \) for some scalar \( k \). Comparing the components: \( 4 = k(-2) \implies k = -2 \) \( 6 = k(-3) \implies k = -2 \) Since the value of \( k \) is consistent (\( k = -2 \)), the vectors are parallel. They point in opposite directions because \( k < 0 \). #### Example 3: Dot Product Calculation Given \( \mathbf{a} = \begin{pmatrix} 5 \\ 1 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} 2 \\ -3 \end{pmatrix} \). Find \( \mathbf{a} \cdot \mathbf{b} \). * **Solution:** \[ \mathbf{a} \cdot \mathbf{b} = (5)(2) + (1)(-3) = 10 - 3 = 7 \] #### Example 4: Finding the Angle Between Two Vectors Find the angle \( \theta \) between \( \mathbf{u} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \) and \( \mathbf{w} = \begin{pmatrix} 3 \\ -1 \end{pmatrix} \). * **Solution:** 1. **Calculate the dot product:** \( \mathbf{u} \cdot \mathbf{w} = (1)(3) + (2)(-1) = 3 - 2 = 1 \) 2. **Calculate the magnitudes:** \( |\mathbf{u}| = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \) \( |\mathbf{w}| = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \) 3. **Use the angle formula:** \[ \cos \theta = \frac{\mathbf{u} \cdot \mathbf{w}}{|\mathbf{u}| |\mathbf{w}|} = \frac{1}{\sqrt{5}\sqrt{10}} = \frac{1}{\sqrt{50}} = \frac{1}{5\sqrt{2}} \] \( \cos \theta \approx 0.1414 \) \( \theta = \cos^{-1}(0.1414) \approx 81.87^\circ \) #### Example 5: Checking for Perpendicular Vectors Are vectors \( \mathbf{r} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \) and \( \mathbf{s} = \begin{pmatrix} -6 \\ 4 \end{pmatrix} \) perpendicular? * **Solution:** Two vectors are perpendicular if their dot product is zero. \[ \mathbf{r} \cdot \mathbf{s} = (2)(-6) + (3)(4) = -12 + 12 = 0 \] Since \( \mathbf{r} \cdot \mathbf{s} = 0 \), the vectors \( \mathbf{r} \) and \( \mathbf{s} \) are perpendicular. ### Exercises 1. Given \( \mathbf{a} = \begin{pmatrix} 4 \\ -1 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} -2 \\ 5 \end{pmatrix} \). * a) Find \( 3\mathbf{a} \). * b) Find \( -\frac{1}{2}\mathbf{b} \). * c) Find \( 2\mathbf{a} + \mathbf{b} \). * d) Determine if \( \mathbf{c} = \begin{pmatrix} 8 \\ -2 \end{pmatrix} \) is parallel to \( \mathbf{a} \). If so, find the scalar \( k \). 2. Calculate the dot product for each pair of vectors: * a) \( \mathbf{p} = \begin{pmatrix} 7 \\ 0 \end{pmatrix} \), \( \mathbf{q} = \begin{pmatrix} -3 \\ 2 \end{pmatrix} \) * b) \( \mathbf{m} = \begin{pmatrix} -4 \\ -2 \end{pmatrix} \), \( \mathbf{n} = \begin{pmatrix} 1 \\ -5 \end{pmatrix} \) 3. Find the angle between the following pairs of vectors: * a) \( \mathbf{u} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \), \( \mathbf{v} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \) * b) \( \mathbf{x} = \begin{pmatrix} 2 \\ 2 \end{pmatrix} \), \( \mathbf{y} = \begin{pmatrix} -3 \\ 3 \end{pmatrix} \) 4. Determine if the vectors \( \mathbf{A} = \begin{pmatrix} 6 \\ -3 \end{pmatrix} \) and \( \mathbf{B} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \) are perpendicular. ### Assignment 1. Given vectors \( \mathbf{u} = \begin{pmatrix} 2 \\ -5 \end{pmatrix} \) and \( \mathbf{v} = \begin{pmatrix} -4 \\ 10 \end{pmatrix} \). * a) Find \( 4\mathbf{u} \). * b) Find \( -\frac{1}{2}\mathbf{v} \). * c) Show that \( \mathbf{u} \) and \( \mathbf{v} \) are parallel vectors. 2. If \( \mathbf{a} = \begin{pmatrix} k \\ 3 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} 4 \\ 6 \end{pmatrix} \) are parallel, find the value of \( k \). 3. If \( \mathbf{p} = \begin{pmatrix} 3 \\ m \end{pmatrix} \) and \( \mathbf{q} = \begin{pmatrix} -2 \\ 4 \end{pmatrix} \) are perpendicular, find the value of \( m \). 4. The position vectors of points P and Q are \( \mathbf{p} = \begin{pmatrix} 1 \\ 7 \end{pmatrix} \) and \( \mathbf{q} = \begin{pmatrix} 5 \\ 3 \end{pmatrix} \) respectively. Find \( \vec{PQ} \) and the magnitude of \( \vec{PQ} \). 5. Two forces \( \mathbf{F_1} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \text{ N} \) and \( \mathbf{F_2} = \begin{pmatrix} -5 \\ 2 \end{pmatrix} \text{ N} \) act on an object. Find the dot product of these two forces and interpret its meaning (what does the sign of the dot product tell you about the angle between the forces?). ### Summary In this lesson, we covered two essential vector operations: * **Scalar Multiplication:** Multiplying a vector by a scalar changes its magnitude and potentially its direction. If \( k > 0 \), the direction remains the same. If \( k < 0 \), the direction reverses. Parallel vectors are scalar multiples of each other. * **Dot Product (Scalar Product):** This operation takes two vectors and results in a scalar. It can be calculated as \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \) or \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \). * The dot product is a powerful tool for determining the angle between vectors. * If the dot product of two non-zero vectors is zero, the vectors are perpendicular. ## Magnitude, direction and sense of a vector ### Introduction Welcome to the exciting world of vectors! In mathematics, we often deal with quantities that have a specific size, like the length of a table or the temperature of a room. These are called **scalar** quantities. However, many real-world situations require us to consider not just the size, but also the way something is moving or acting. For example, if you push a box, you need to know how hard you're pushing (the force) and in what direction. This is where **vectors** come in. Vectors are mathematical tools that help us describe quantities that have both a size (magnitude) and a specific direction. In this lesson, you will learn what vectors are, how to represent them, and how to understand their key characteristics: magnitude, direction, and sense. ### Key Concepts #### Scalar vs. Vector Quantities * **Scalar Quantity:** A quantity that has only magnitude (size). It does not have a direction. * Examples: Mass, time, temperature, distance, speed. * **Vector Quantity:** A quantity that has both magnitude and direction. * Examples: Force, velocity, displacement, acceleration. #### Representation of a Vector Vectors can be represented in several ways: * **Geometrically:** A directed line segment (an arrow). The length of the arrow represents the magnitude, and the arrowhead indicates the direction and sense. * If a vector starts at point A and ends at point B, it is denoted as \(\vec{AB}\). * **Symbolically:** Using a single bold lowercase letter (e.g., \(\mathbf{a}\), \(\mathbf{b}\)) or a lowercase letter with an arrow above it (e.g., \(\vec{a}\), \(\vec{b}\)). * **Column Vector (Component Form):** In a 2-dimensional coordinate system, a vector can be written as a column matrix showing its horizontal (x) and vertical (y) components. * A vector \(\mathbf{v}\) with x-component \(x\) and y-component \(y\) is written as: \[ \mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix} \] #### Magnitude of a Vector The **magnitude** of a vector is its length or size. It tells us 'how much' of the quantity the vector represents. * **Notation:** The magnitude of vector \(\mathbf{a}\) is denoted as \(|\mathbf{a}|\) or \(|\vec{a}|\). * **Calculation:** If a vector \(\mathbf{v}\) is given in column form as \(\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}\), its magnitude can be found using the Pythagorean theorem: \[ |\mathbf{v}| = \sqrt{x^2 + y^2} \] * **Vector between two points:** If a vector \(\vec{AB}\) starts at point \(A(x_1, y_1)\) and ends at point \(B(x_2, y_2)\), its column form is: \[ \vec{AB} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix} \] And its magnitude is: \[ |\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] #### Direction of a Vector The **direction** of a vector refers to the line along which the vector acts. For example, a vector could be acting along a horizontal line, a vertical line, or a diagonal line. * It defines the orientation of the vector in space. #### Sense of a Vector The **sense** of a vector refers to the specific way the vector is pointing along its line of direction. It is indicated by the arrowhead. * For example, if a vector is acting along a horizontal line, its sense could be 'to the right' or 'to the left'. * Vectors \(\vec{AB}\) and \(\vec{BA}\) have the same magnitude and lie along the same direction, but they have opposite senses. #### Position Vector A **position vector** is a vector that starts from the origin \((0,0)\) and ends at a specific point. * If point \(P\) has coordinates \((x,y)\), its position vector is \(\vec{OP} = \begin{pmatrix} x \\ y \end{pmatrix}\). ### Vocabulary * **Scalar:** A physical quantity that has magnitude but no direction. * **Vector:** A physical quantity that has both magnitude and direction. * **Magnitude:** The length or size of a vector, representing its value. * **Direction:** The line along which a vector acts. * **Sense:** The specific way a vector points along its line of direction, indicated by the arrowhead. * **Position Vector:** A vector that starts from the origin \((0,0)\) and points to a specific point. * **Column Vector:** A representation of a vector using its x and y components, written as \(\begin{pmatrix} x \\ y \end{pmatrix}\). * **Origin:** The point \((0,0)\) on a coordinate plane. ### Examples #### Example 1: Finding the Magnitude of a Position Vector Given the position vector \(\mathbf{p} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\), find its magnitude. **Solution:** Using the formula \(|\mathbf{p}| = \sqrt{x^2 + y^2}\): \[ |\mathbf{p}| = \sqrt{3^2 + 4^2} \] \[ |\mathbf{p}| = \sqrt{9 + 16} \] \[ |\mathbf{p}| = \sqrt{25} \] \[ |\mathbf{p}| = 5 \] The magnitude of vector \(\mathbf{p}\) is 5 units. #### Example 2: Finding a Vector Between Two Points Find the vector \(\vec{AB}\) given point \(A(1, 2)\) and point \(B(5, -1)\). **Solution:** Using the formula \(\vec{AB} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix}\): \[ \vec{AB} = \begin{pmatrix} 5 - 1 \\ -1 - 2 \end{pmatrix} \] \[ \vec{AB} = \begin{pmatrix} 4 \\ -3 \end{pmatrix} \] The vector \(\vec{AB}\) is \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\). #### Example 3: Finding the Magnitude of a Vector Between Two Points Using the vector \(\vec{AB}\) from Example 2, find its magnitude. **Solution:** From Example 2, \(\vec{AB} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}\). Using the magnitude formula: \[ |\vec{AB}| = \sqrt{4^2 + (-3)^2} \] \[ |\vec{AB}| = \sqrt{16 + 9} \] \[ |\vec{AB}| = \sqrt{25} \] \[ |\vec{AB}| = 5 \] The magnitude of vector \(\vec{AB}\) is 5 units. #### Example 4: Understanding Direction and Sense Consider vector \(\vec{PQ}\) from \(P(0,0)\) to \(Q(2,3)\) and vector \(\vec{QP}\) from \(Q(2,3)\) to \(P(0,0)\). * **Direction:** Both vectors \(\vec{PQ}\) and \(\vec{QP}\) lie on the same straight line passing through points \(P\) and \(Q\). Therefore, they have the same direction. * **Sense:** Vector \(\vec{PQ}\) points from \(P\) towards \(Q\), while vector \(\vec{QP}\) points from \(Q\) towards \(P\). They point in opposite ways along the same line. Thus, they have opposite senses. ### Exercises 1. Represent the vector \(\vec{PQ}\) in column form given point \(P(2, 5)\) and point \(Q(7, 1)\). 2. Calculate the magnitude of the vector \(\mathbf{a} = \begin{pmatrix} -6 \\ 8 \end{pmatrix}\). 3. A vector \(\mathbf{b}\) starts at point \((3, 0)\) and ends at point \((0, 4)\). a) Write \(\mathbf{b}\) in column form. b) Find the magnitude of \(\mathbf{b}\). 4. True or False: If two vectors have the same magnitude, they must have the same direction. Explain your answer. 5. Describe in your own words the difference between the 'direction' and the 'sense' of a vector. ### Assignment 1. Given points \(A(-1, 3)\), \(B(4, 3)\), \(C(4, -2)\), and \(D(-1, -2)\). a) Write the column vectors for \(\vec{AB}\), \(\vec{BC}\), \(\vec{CD}\), and \(\vec{DA}\). b) Find the magnitudes of each of these vectors. c) What do you notice about the magnitudes of these vectors? What geometric shape do these points form when connected in order? 2. A force vector \(\mathbf{F} = \begin{pmatrix} 8 \\ -6 \end{pmatrix}\) N (Newtons) is applied to an object. a) What is the magnitude of the force? b) If another force \(\mathbf{G} = \begin{pmatrix} -8 \\ 6 \end{pmatrix}\) N is applied, compare the magnitude, direction, and sense of \(\mathbf{F}\) and \(\mathbf{G}\). 3. Explain why 'displacement' is a vector quantity, while 'distance' is a scalar quantity. Provide an example to illustrate your explanation. 4. A ship sails from port P to port Q. The position vector of P is \(\begin{pmatrix} 2 \\ 5 \end{pmatrix}\) and the position vector of Q is \(\begin{pmatrix} 8 \\ 1 \end{pmatrix}\). a) Find the column vector representing the displacement from P to Q (i.e., \(\vec{PQ}\)). b) Calculate the distance between P and Q (this is the magnitude of the displacement vector). ### Summary In this lesson, we explored the fundamental concepts of vectors. We learned that: * **Scalars** have only magnitude, while **vectors** have both magnitude and direction. * Vectors can be represented geometrically as arrows or symbolically as column vectors like \(\begin{pmatrix} x \\ y \end{pmatrix}\). * The **magnitude** of a vector is its length, calculated using the formula \(|\mathbf{v}| = \sqrt{x^2 + y^2}\). * The **direction** of a vector is the line along which it acts. * The **sense** of a vector is the specific way it points along its direction, indicated by the arrowhead. * A **position vector** starts from the origin \((0,0)\) and points to a given coordinate.