Question

If you have three unit vectors such that two of them are along the coordinate axis, is it possible for the resultant to be a unit vector?

• A. Yes
• B. No

Answer 1

The correct option is A

Yes

The magnitude of the resultant of unit vectors along the axes will be $\sqrt{2}$

E.g.:

magnitude of $\hat{i}+\hat{j}$ is $\sqrt{2}$

Let $|\overrightarrow{R}|$ be the resultant and $\overrightarrow{A}$ and $\overrightarrow{B}$ be unit vectors along the axes,$|\overrightarrow{C}|$ is the third unit vector.

$\therefore |\overrightarrow{R}{|}^2=|\overrightarrow{A}+\overrightarrow{B}{|}^2+|\overrightarrow{C}{|}^2+2|\overrightarrow{A}+\overrightarrow{B}||\overrightarrow{C}|cos\theta$

Where $\theta$ is the angle between $\overrightarrow{A}+\overrightarrow{B}$ and $\overrightarrow{C}$.

$1^2=(\sqrt{2}{)}^2+1^2+2\sqrt{2}cos\theta$

$1=2+1+2\sqrt{2}cos\theta$

$\Rightarrow 1=3+2\sqrt{2}cos\theta$

$\Rightarrow -2=2\sqrt{2}cos\theta$

$\Rightarrow cos\theta =\frac{-1}{\sqrt{2}}\Rightarrow ={135}^{\circ }$ $(cos{135}^{\circ }=\frac{-1}{\sqrt{2}})$

Since it produces a valid solution, we can see that it is surely possible and also that angle $\overrightarrow{A}+\overrightarrow{B}$ and $\overrightarrow{C}$ should be ${135}^{\circ }$.Let us take an example, Let $\overrightarrow{A}$ be $\hat{i}$,$\overrightarrow{B}$ be $\hat{j}$, then $\overrightarrow{C}$ will be $-\hat{i}$.

We can also see that $\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}=\hat{i}+\hat{j}+-\hat{i}=\hat{j}$ which is a unit vector