3.1 Q-4

Exercise 3.1 — Question 4

Problem

Find the vector $\overrightarrow{PQ}$ and its magnitude, given the points $P$ and $Q$ in 3D space. Also determine whether the given vectors are parallel.

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Key Concepts

Vector Between Two Points

Given two points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ in 3D space, the vector from $P$ to $Q$ is:

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

where $O$ is the origin.

Magnitude of a Vector

For a vector $\vec{v}=a\hat{i}+b\hat{j}+c\hat{k}$, the magnitude is:

$|\vec{v}|=\sqrt{a^2+b^2+c^2}$

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:

$\mathbf{a}=k\mathbf{b}\text{for some scalar }k\ne 0$

In component form:

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

• If $k>0$: vectors point in the same direction
• If $k<0$: vectors are antiparallel (opposite directions)

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Method

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

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

Step 3: Find the magnitude: $|\overrightarrow{PQ}|=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$

Step 4: To check if two vectors are parallel, verify whether their components are proportional.

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