3.1 Q-2

Exercise 3.1 — Question 2

This exercise covers fundamental vector operations in 3D space, including finding vectors between two points, checking for parallel vectors, and applying basic vector algebra.

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Key Concepts

Rectangular Coordinate System in Space

In 3D space, every point is located using three mutually perpendicular axes: the $x$-axis, $y$-axis, and $z$-axis. A point $P$ is written as an ordered triple $P(x,y,z)$.

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

The standard unit vectors along the positive $x$-, $y$-, and $z$-axes are: $\hat{i}=(1,0,0),\hat{j}=(0,1,0),\hat{k}=(0,0,1)$

Any vector $\vec{v}$ in space can be written as: $\vec{v}=v_1\hat{i}+v_2\hat{j}+v_3\hat{k}$ where $v_1$, $v_2$, $v_3$ are the components of $\vec{v}$.

Vector Between Two Points

Given points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$, the vector from $P$ to $Q$ is: $\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}$

Magnitude of a Vector

The magnitude (length) of $\vec{v}=v_1\hat{i}+v_2\hat{j}+v_3\hat{k}$ is: $|\vec{v}|=\sqrt{v_1^2+v_2^2+v_3^2}$

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

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Worked Examples

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

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

Example 2: Are $\mathbf{a}=2\hat{i}-4\hat{j}+6\hat{k}$ and $\mathbf{b}=\hat{i}-2\hat{j}+3\hat{k}$ parallel?

$\frac{2}{1}=\frac{-4}{-2}=\frac{6}{3}=2$

Since all ratios are equal ($k=2$), the vectors are parallel.

Example 3: Find the magnitude of $\overrightarrow{PQ}=3\hat{i}+2\hat{j}-4\hat{k}$.

$|\overrightarrow{PQ}|=\sqrt{3^2+2^2+(-4{)}^2}=\sqrt{9+4+16}=\sqrt{29}$

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