8.1 Vectors
In mathematics and physics, quantities are divided into two main categories:
- Scalar Quantity: A quantity that has magnitude only (e.g., mass, distance, speed, area, time, temperature).
- Vector Quantity: A quantity that has both magnitude and direction (e.g., displacement, velocity, acceleration, force, momentum).
Representation of Vectors
A vector can be represented by a directed line segment starting from an initial point $A$ to a terminal point $B$:
- Notation: Represented as $\vec{AB}$ or a bold symbol $\mathbf{a}$ or underlined symbol $\underline{a}$.
- Magnitude: The length of the line segment represents the magnitude, denoted as $|\vec{AB}|$ or $|\mathbf{a}|$. Magnitude is always non-negative: $|\mathbf{a}| \ge 0$.
Negative Vectors and Zero Vector
- Negative Vector: The negative vector of $\vec{AB}$ is $\vec{BA}$, where $\vec{BA} = -\vec{AB}$. It has the same magnitude but operates in the opposite direction.
- Zero Vector ($\mathbf{0}$): A vector with zero magnitude and no specific direction. $|\mathbf{0}| = 0$. Example: $\vec{AA} = \mathbf{0}$.
Equal Vectors
Two vectors $\mathbf{a}$ and $\mathbf{b}$ are equal ($\mathbf{a} = \mathbf{b}$) if and only if they have:
- The same magnitude ($|\mathbf{a}| = |\mathbf{b}|$).
- The same direction.
Parallel Vectors and Collinearity
- Two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ are parallel ($\mathbf{a} \parallel \mathbf{b}$) if and only if:
$$\mathbf{a} = k\mathbf{b}$$
where $k$ is a non-zero scalar constant.
- If $k > 0$, $\mathbf{a}$ and $\mathbf{b}$ have the same direction.
- If $k < 0$, $\mathbf{a}$ and $\mathbf{b}$ have opposite directions.
- Collinear Points: Points $A$, $B$, and $C$ lie on a straight line (are collinear) if:
$$\vec{AB} = k\vec{BC} \quad \text{or} \quad \vec{AB} = m\vec{AC}$$
and share a common point ($B$ or $A$).
8.2 Addition and Subtraction of Vectors
Scalar Multiplication of Vectors
When a vector $\mathbf{a}$ is multiplied by a scalar $k$:
- The magnitude becomes $|k| \times |\mathbf{a}|$.
- If $k > 0$, direction remains identical to $\mathbf{a}$.
- If $k < 0$, direction reverses relative to $\mathbf{a}$.
Addition of Non-Parallel Vectors (Resultant Vector)
The sum of two or more vectors is called the resultant vector.
- Triangle Law: To find $\mathbf{a} + \mathbf{b}$, place the tail of $\mathbf{b}$ at the head of $\mathbf{a}$. The resultant vector $\vec{AC} = \vec{AB} + \vec{BC}$.
- Parallelogram Law: If vectors $\vec{AB} = \mathbf{a}$ and $\vec{AD} = \mathbf{b}$ originate from the same initial point $A$, complete the parallelogram $ABCD$. The diagonal $\vec{AC} = \mathbf{a} + \mathbf{b}$.
- Polygon Law: For adding three or more vectors sequentially: $\vec{AE} = \vec{AB} + \vec{BC} + \vec{CD} + \vec{DE}$.
Subtraction of Vectors
Subtraction of vector $\mathbf{b}$ from $\mathbf{a}$ is defined as adding the negative vector of $\mathbf{b}$:
$$\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b})$$
8.3 Vectors in the Cartesian Plane
Representation of Vectors in Component Form
In a 2D Cartesian plane, a vector $\mathbf{r}$ can be expressed using unit vectors $\mathbf{i}$ and $\mathbf{j}$ along the $x$-axis and $y$-axis respectively:
- Unit Vectors: $\mathbf{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\mathbf{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$.
- Linear Combination Form: $\mathbf{r} = x\mathbf{i} + y\mathbf{j}$
- Column Vector Form: $\mathbf{r} = \begin{pmatrix} x \\ y \end{pmatrix}$
Magnitude of a Vector in Cartesian Plane
Using the Pythagorean Theorem, the magnitude of vector $\mathbf{r} = x\mathbf{i} + y\mathbf{j} = \begin{pmatrix} x \\ y \end{pmatrix}$ is:
$$|\mathbf{r}| = \sqrt{x^2 + y^2}$$
Unit Vector in the Direction of a Vector
A unit vector is a vector with a magnitude of 1 unit. The unit vector in the direction of $\mathbf{r}$, denoted as $\hat{\mathbf{r}}$ (read as 'r-hat'), is given by:
$$\hat{\mathbf{r}} = \frac{\mathbf{r}}{|\mathbf{r}|} = \frac{x\mathbf{i} + y\mathbf{j}}{\sqrt{x^2 + y^2}} = \begin{pmatrix} \frac{x}{\sqrt{x^2 + y^2}} \\ \frac{y}{\sqrt{x^2 + y^2}} \end{pmatrix}$$
Arithmetic Operations on Component Vectors
Given $\mathbf{a} = x_1\mathbf{i} + y_1\mathbf{j} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}$ and $\mathbf{b} = x_2\mathbf{i} + y_2\mathbf{j} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}$:
- Addition: $\mathbf{a} + \mathbf{b} = (x_1 + x_2)\mathbf{i} + (y_1 + y_2)\mathbf{j} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}$
- Subtraction: $\mathbf{a} - \mathbf{b} = (x_1 - x_2)\mathbf{i} + (y_1 - y_2)\mathbf{j} = \begin{pmatrix} x_1 - x_2 \\ y_1 - y_2 \end{pmatrix}$
- Scalar Multiplication: $k\mathbf{a} = k(x_1\mathbf{i} + y_1\mathbf{j}) = (kx_1)\mathbf{i} + (ky_1)\mathbf{j} = \begin{pmatrix} kx_1 \\ ky_1 \end{pmatrix}$
Position Vectors
If $O(0,0)$ is the origin and $P(x,y)$ is any point in the Cartesian plane, the position vector of point $P$ relative to origin $O$ is:
$$\vec{OP} = \mathbf{p} = x\mathbf{i} + y\mathbf{j} = \begin{pmatrix} x \\ y \end{pmatrix}$$
For any two points $A(x_1, y_1)$ and $B(x_2, y_2)$ with position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively:
$$\vec{AB} = \vec{AO} + \vec{OB} = -\vec{OA} + \vec{OB} = \mathbf{b} - \mathbf{a} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix}$$