Q10DQ - University Physics with Modern Physics

Question

Is the vector $(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ a unit vector? Is the vector $(3.0\hat{\mathrm{i}}-2.0\hat{\mathrm{j}})$ a unit vector? Justify your answers.

Step-by-Step Solution

Answer

The vectors $(\hat{i}+\hat{j}+\hat{k})$ and $(3.0\hat{i}-2.0\hat{j})$ are not a unit vector.

Introduction

Unit vectors are the vectors whose magnitude is exactly 1 unit. They are very useful for various reasons. Specifically, the unit vectors $[0,1]$ and $[1,0]$ can together form any other vector.

Explanation

Unit Vector - A unit vector is a vector with magnitude 1.

Let the vector $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$.

Magnitude:

$$\begin{gathered} \left|\overrightarrow{a}\right|=\sqrt{1^2+1^2+1^2} \\ =\sqrt{3} \end{gathered}$$

Since magnitude is not $1,\overrightarrow{a}$ is not a unit vector.

Likewise, assume the vector $\overrightarrow{b}=3.0\hat{i}-2.0\hat{j}$.

The amplitude of the vector is

$\begin{array}{rcl}\left|\overrightarrow{b}\right|& =& \sqrt{3^2+2^2}\\ & =& \sqrt{9+4}\\ & =& \sqrt{13}\end{array}$

Which is not equal to 1 .

Hence, it is not a unit vector.