Q.Find the scalar and vector components of the vector with initial point $(2, 1)$ and terminal point $(-5, 7)$.

Concept: Vector Component Extraction — the vector from $P$ to $Q$ is $Q - P$.

Step 1

Let initial point $A = (2, 1)$ and terminal point $B = (-5, 7)$.

The vector $\overrightarrow{AB} = B - A$.

Step 2

Compute components:

$x$-component: $-5 - 2 = -7$

$y$-component: $7 - 1 = 6$

Step 3

Scalar components are these individual numbers: $-7$ and $6$. …

Why this works: Vector Component Extraction

A vector is defined by its displacement — how far it moves in each direction. The initial point tells you where you start; the terminal point tells you where you end. The vector itself is simply the difference: end minus start.

The scalar components are just the numbers that multiply the unit vectors $\hat{i}$ and $\hat{j}$. They tell you how many steps you take along each axis. The vector components are those scalar components attached to their unit vectors — the actual pieces that add up to the full vector.

So the entire problem reduces to one subtraction: terminal coordinates minus initial coordinates.

Step-by-step solution

1. Identify the coordinates.

Let the initial point be $A(2, 1)$ and the terminal point be $B(-5, 7)$. The vector is $\overrightarrow{AB}$.

2. Find the displacement in $x$.

The $x$-coordinate changes from $2$ to $-5$.

This $-7$ is the scalar component along $\hat{i}$. It means the vector points 7 units in the negative $x$-direction.

3. Find the displacement in $y$.

The $y$-coordinate changes from $1$ to $7$.

This $6$ is the scalar component along $\hat{j}$. The vector points 6 units in the positive $y$-direction.

4. Write the vector in component form.

$\overrightarrow{AB} = (-7)\hat{i} + (6)\hat{j}$ …

Method: Scalar and vector components of a directed segment

Use this whenever a vector is given by an initial point $P$ and terminal point $Q$.

Steps

Step 1: Form the vector as terminal minus initial

Always subtract initial from terminal, not the reverse.

Step 2: Read off the scalar components

The scalar components are the plain numbers multiplying $\hat{i}$ and $\hat{j}$ — here $x_2-x_1$ and $y_2-y_1$ (they may be negative).

Step 3: Write the vector components …

Common Mistakes

Mistake 1: Subtracting in the wrong order (initial minus terminal)

Why it's wrong: $P-Q$ gives the opposite vector, flipping every sign. Correct approach: always compute $Q-P=(-5-2)\hat i+(7-1)\hat j=-7\hat i+6\hat j$.

Mistake 2: Confusing scalar components with vector components …