Vector Geometry Classroom

49.1 Introduction

Vector geometry is a branch of mathematics that deals with vectors, which are quantities that have both magnitude (size) and direction. Unlike scalars (which only have magnitude), vectors are essential for describing physical phenomena such as displacement, velocity, force, and acceleration. It provides a powerful framework for understanding motion, forces, and spatial relationships in two and three dimensions.

In this classroom, you will explore the fundamental concepts of vectors, including their representation, operations (addition, subtraction, resolution), and applications in various fields.

49.2 Scalars & Vectors

In physics and mathematics, quantities are often categorized as either scalars or vectors:

• Scalar Quantity: A quantity that has only magnitude (size). It is fully described by a numerical value and a unit. Examples include temperature, mass, time, speed, and distance.
• Vector Quantity: A quantity that has both magnitude and direction. It requires both a numerical value (magnitude) and a specified direction to be fully described. Examples include displacement, velocity, force, acceleration, and momentum.

Understanding the distinction between scalars and vectors is crucial for correctly modeling and solving problems in physics and engineering.

49.3 Drawing Vectors

Vectors are typically represented graphically by arrows. The length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector. The starting point of the arrow is called the tail, and the tip is called the head.

When drawing vectors to scale, it's important to choose a convenient scale (e.g., $1 \text{ cm} = 10 \text{ N}$ for force) and use a protractor to accurately represent the direction. A vector can be represented by a single letter with an arrow above it, like $\vec{v}$, or in bold, like $\mathbf{v}$.

49.4 Vector Addition (Drawing)

Graphical vector addition can be performed using two main methods: the head-to-tail method and the parallelogram method.

• Head-to-Tail Method: To add vector $\vec{A}$ and vector $\vec{B}$, draw $\vec{A}$. Then, draw $\vec{B}$ starting from the head of $\vec{A}$. The resultant vector, $\vec{R} = \vec{A} + \vec{B}$, is drawn from the tail of $\vec{A}$ to the head of $\vec{B}$.
• Parallelogram Method: To add vector $\vec{A}$ and vector $\vec{B}$, draw both vectors originating from the same point. Complete the parallelogram using these two vectors as adjacent sides. The resultant vector is the diagonal drawn from the common origin to the opposite vertex of the parallelogram.

Both methods yield the same resultant vector.

49.5 Resolving Vectors

Resolving a vector means breaking it down into its component vectors, usually along perpendicular axes (e.g., horizontal and vertical axes in a Cartesian coordinate system). If a vector $\vec{V}$ has magnitude $V$ and makes an angle $\theta$ with the positive x-axis, its components are:

• Horizontal component ($V_x$): $V_x = V \cos \theta$
• Vertical component ($V_y$): $V_y = V \sin \theta$

This process is crucial for vector addition and subtraction by calculation, as it allows you to combine vectors by simply adding or subtracting their corresponding components.

49.6 Vector Addition (Calculation)

To add vectors numerically, especially when they are expressed in terms of components or angles, follow these steps:

1. Resolve each vector into its horizontal (x) and vertical (y) components.
2. Add all x-components together to find the resultant x-component ($\Sigma V_x$).
3. Add all y-components together to find the resultant y-component ($\Sigma V_y$).
4. Calculate the magnitude of the resultant vector $R$ using the Pythagorean theorem: $R = \sqrt{(\Sigma V_x)^2 + (\Sigma V_y)^2}$.
5. Calculate the direction of the resultant vector using trigonometry: $\tan \theta = \frac{\Sigma V_y}{\Sigma V_x}$, ensuring to consider the quadrant of the resultant components for the correct angle.

This method is more precise and efficient than graphical methods for complex problems.

49.7 Vector Subtraction

Vector subtraction, $\vec{A} - \vec{B}$, can be thought of as adding the negative of vector $\vec{B}$ to vector $\vec{A}$: $\vec{A} + (-\vec{B})$. The negative of a vector has the same magnitude but the opposite direction.

Numerically, if $\vec{A} = A_x \mathbf{i} + A_y \mathbf{j}$ and $\vec{B} = B_x \mathbf{i} + B_y \mathbf{j}$, then:

Graphically, draw $\vec{A}$ and then draw $-\vec{B}$ (same length as $\vec{B}$ but pointing in the opposite direction) starting from the head of $\vec{A}$. The resultant is from the tail of $\vec{A}$ to the head of $-\vec{B}$.

49.8 Relative Velocity

Relative velocity is the velocity of an object or observer $A$ relative to another object or observer $B$. It is given by the vector difference between their velocities:

Where $\vec{v}_{A/B}$ is the velocity of A relative to B, $\vec{v}_A$ is the velocity of A relative to a fixed reference point (e.g., the ground), and $\vec{v}_B$ is the velocity of B relative to the same fixed reference point.

Relative velocity problems often involve combining vector addition and subtraction principles, especially in scenarios with motion in two dimensions (e.g., airplanes in wind, boats in currents).

49.9 i, j, k Notation

Vectors in two and three dimensions can be conveniently expressed using unit vectors along the coordinate axes. These are typically denoted as:

• $\mathbf{i}$ (or $\hat{i}$): A unit vector in the direction of the positive x-axis.
• $\mathbf{j}$ (or $\hat{j}$): A unit vector in the direction of the positive y-axis.
• $\mathbf{k}$ (or $\hat{k}$): A unit vector in the direction of the positive z-axis (for 3D vectors).

A vector $\vec{V}$ with components $(V_x, V_y)$ in 2D can be written as $\vec{V} = V_x \mathbf{i} + V_y \mathbf{j}$. In 3D, $\vec{V} = V_x \mathbf{i} + V_y \mathbf{j} + V_z \mathbf{k}$. This notation simplifies algebraic operations with vectors.

51.1 Unit Triad

The unit triad in three-dimensional Cartesian coordinates consists of three mutually perpendicular unit vectors, conventionally denoted as $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$. These vectors define the directions of the positive x, y, and z axes, respectively.

• $\mathbf{i} = \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}$
• $\mathbf{j} = \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}$
• $\mathbf{k} = \begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix}$

Each unit vector has a magnitude of 1. Any vector in 3D space can be expressed as a linear combination of these three unit vectors, making them fundamental for vector operations and transformations.

51.2 Scalar Product (Dot Product)

The scalar product, also known as the dot product, of two vectors $\vec{A}$ and $\vec{B}$ is a scalar quantity defined as:

where $|\vec{A}|$ and $|\vec{B}|$ are the magnitudes of the vectors, and $\theta$ is the angle between them. In Cartesian coordinates, if $\vec{A} = A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k}$ and $\vec{B} = B_x \mathbf{i} + B_y \mathbf{j} + B_z \mathbf{k}$, the dot product is calculated as:

The scalar product is used to find the angle between two vectors and to determine if two vectors are perpendicular (if their dot product is zero).

51.3 Vector Products (Cross Product)

The vector product, also known as the cross product, of two vectors $\vec{A}$ and $\vec{B}$ is a vector quantity defined as:

where $\hat{n}$ is a unit vector perpendicular to both $\vec{A}$ and $\vec{B}$ in the direction given by the right-hand rule. The magnitude $|\vec{A} \times \vec{B}|$ represents the area of the parallelogram formed by $\vec{A}$ and $\vec{B}$. In Cartesian coordinates, if $\vec{A} = A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k}$ and $\vec{B} = B_x \mathbf{i} + B_y \mathbf{j} + B_z \mathbf{k}$, the cross product is calculated as:

The vector product is used to find a vector perpendicular to two given vectors and in calculations involving torque and angular momentum.

51.4 Vector Equation of a Line

The vector equation of a line in 2D or 3D space defines all points on the line. It is typically given by:

where:

• $\vec{r}$ is the position vector of any point $(x, y, z)$ on the line.
• $\vec{a}$ is the position vector of a known point $(a_x, a_y, a_z)$ on the line.
• $\vec{d}$ is a direction vector $(d_x, d_y, d_z)$ parallel to the line.
• $t$ is a scalar parameter that can take any real value.

This equation provides a concise way to represent a line and is fundamental in various geometry and physics problems, including trajectory analysis and intersections of lines and planes.