3.1 Q-9

Exercise 3.1 — Question 9

Problem Statement

Find the vector $\overrightarrow{PQ}$ given two points $P$ and $Q$, and determine whether the given vectors are parallel.

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Key Concepts

Finding a Vector from Two Points

Given two points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ in space, the vector $\overrightarrow{PQ}$ is found by subtracting the position vector of the initial point from the position vector of the terminal point:

$\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}$

In component form:

$\overrightarrow{PQ}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}$

For 2D points $P(x_1,y_1)$ and $Q(x_2,y_2)$:

$\overrightarrow{PQ}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}$

Condition for Parallel Vectors

Two vectors $\mathbf{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\mathbf{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ are parallel if and only if one is a scalar multiple of the other:

$\mathbf{a}=k\mathbf{b}\text{for some scalar }k$

In component form, this means their components are proportional:

$\frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}$

In 2D, the equivalent algebraic condition is:

$a_1{b}_2-a_2{b}_1=0$

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Worked Solution

Step 1: Identify the coordinates of the given points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$.

Step 2: Apply the formula: $\overrightarrow{PQ}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}$

Step 3: To check if two vectors are parallel, verify that their components are proportional: $\frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}$

If this condition holds, the vectors are parallel. If the ratios are not all equal, the vectors are not parallel.

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