Exercise 3.1 — Question 10

Problem Statement

This question involves applying fundamental vector operations in 3D space, including finding vectors between two points, computing magnitudes, and verifying properties such as parallelism or collinearity.

---

Key Concepts

Vector Between Two Points

Given two points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$, the vector $\overrightarrow{PQ}$ 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:

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

Unit Vectors $\hat{i}$, $\hat{j}$, $\hat{k}$

• $\hat{i}$ — unit vector along the positive $x$-axis
• $\hat{j}$ — unit vector along the positive $y$-axis
• $\hat{k}$ — unit vector along the positive $z$-axis

Each has magnitude 1: $|\hat{i}|=|\hat{j}|=|\hat{k}|=1$

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 one is a scalar multiple of the other:

$\mathbf{a}=k\mathbf{b}⟺\frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}$

Properties of Vector Addition

---

Worked Example

Given: Points $A(1,2,3)$ and $B(4,6,8)$.

Find: $\overrightarrow{AB}$ and $|\overrightarrow{AB}|$.

Solution:

$\overrightarrow{AB}=(4-1)\hat{i}+(6-2)\hat{j}+(8-3)\hat{k}=3\hat{i}+4\hat{j}+5\hat{k}$

$|\overrightarrow{AB}|=\sqrt{3^2+4^2+5^2}=\sqrt{9+16+25}=\sqrt{50}=5\sqrt{2}$

---