Exercise 3.1 — Question 5

Concepts Required

This question draws on vectors in 3D space, including:

• The rectangular coordinate system in space (x, y, z axes)
• Unit vectors $\hat{i}$, $\hat{j}$, $\hat{k}$ along the x, y, z axes respectively
• Vector between two points: $\overrightarrow{PQ}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}$
• Magnitude of a vector: $|\vec{v}|=\sqrt{v_1^2+v_2^2+v_3^2}$
• Parallel vectors: $\mathbf{a}\parallel \mathbf{b}$ iff $\frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}$

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Key Formula: Vector Between Two Points

Given points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$:

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

Note that $\overrightarrow{QP}=-\overrightarrow{PQ}$, illustrating the additive inverse property of vectors.

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Condition for Parallel Vectors

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:

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

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Properties of Vector Addition (Revision)

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Worked Example

Given: $P(1,2,-1)$ and $Q(3,0,4)$. Find $\overrightarrow{PQ}$, $\overrightarrow{QP}$, and $|\overrightarrow{PQ}|$.

Step 1: Find $\overrightarrow{PQ}$: $\overrightarrow{PQ}=(3-1)\hat{i}+(0-2)\hat{j}+(4-(-1))\hat{k}=2\hat{i}-2\hat{j}+5\hat{k}$

Step 2: Find $\overrightarrow{QP}$: $\overrightarrow{QP}=-\overrightarrow{PQ}=-2\hat{i}+2\hat{j}-5\hat{k}$

Step 3: Find $|\overrightarrow{PQ}|$: $|\overrightarrow{PQ}|=\sqrt{(2{)}^2+(-2{)}^2+(5{)}^2}=\sqrt{4+4+25}=\sqrt{33}$

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