Question

If $\overrightarrow{A}=2\hat{i}-2\hat{j}-\hat{k}$ and $\overrightarrow{B}=\hat{i}+\hat{j}$, then
(i) find the angle between $\overrightarrow{A}$ and $\overrightarrow{B}$
(ii) find the projection of resultant vector of $\overrightarrow{A}$ and $\overrightarrow{B}$ on x-axis.
(iii) find a vector which is, if added to $\overrightarrow{A}$, gives a unit vector along y-axis.

• A. $\theta ={90}^{\circ }$; $3$; $-2\hat{i}+3\hat{j}+\hat{k}$
• B. $\theta ={90}^{\circ }$; $5$; $-2\hat{i}+4\hat{j}+\hat{k}$
• C. $\theta ={45}^{\circ }$; $3$; $-3\hat{i}+3\hat{j}+\hat{k}$
• D. $\theta ={60}^{\circ }$; $3$; $-2\hat{i}+3\hat{j}+\hat{k}$

Answer 1

The correct option is A $\theta ={90}^{\circ }$; $3$; $-2\hat{i}+3\hat{j}+\hat{k}$

i) $\overrightarrow{A}.\overrightarrow{B}=AB\cos \theta$

or $(2\hat{i}-2\hat{j}-\hat{k}).(\hat{i}+\hat{j})=AB\cos \theta$

or $2-2=AB\cos \theta$ or $cos\theta =0\Rightarrow \theta ={90}^o$

ii) Resultant, $\overrightarrow{R}=\overrightarrow{A}+\overrightarrow{B}=(2\hat{i}-2\hat{j}-\hat{k})+(\hat{i}+\hat{j})=3\hat{i}-\hat{j}-\hat{k}$

The projection of resultant on x -axis $=\overrightarrow{R}.\hat{i}=3$

iii) Here, $\overrightarrow{P}+\overrightarrow{A}=\hat{j}$

$\Rightarrow \overrightarrow{P}=\hat{j}-\overrightarrow{A}=\hat{j}-(2\hat{i}-2\hat{j}-\hat{k})=-2\hat{i}+3\hat{j}+\hat{k}$