Question

The projection of the vector $2\hat{i}+\hat{j}-3\hat{k}$ on the vector $\hat{i}-2\hat{j}-\hat{k}$

• A. $-\frac{3}{\sqrt{14}}$
• B. $\frac{3}{\sqrt{14}}$
• C. $-\sqrt{\frac{3}{2}}$
• D. $\sqrt{\frac{3}{2}}$

Answer 1

The correct option is D $\sqrt{\frac{3}{2}}$

Given that:

$\overrightarrow{P}=2\hat{i}+\hat{j}-3\hat{k}$

$\overrightarrow{Q}=\hat{i}-2\hat{j}-\hat{k}$

Now

We know that projection of $\overrightarrow{P}$ on $\overrightarrow{Q}$ is :$\frac{\overrightarrow{P}.\overrightarrow{Q}}{|\overrightarrow{Q}|}$

So,

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

$\overrightarrow{P}.\overrightarrow{Q}=2\times 1+(1\times (-2\left)\right)+(-3\times -1)$

$\overrightarrow{P}.\overrightarrow{Q}=2-2+3$

$\overrightarrow{P}.\overrightarrow{Q}=3$

Now

$|\overrightarrow{Q}|=\sqrt{1+4+1}=\sqrt{6}$

So,

$proj\overrightarrow{Q}\overrightarrow{P}=\frac{3}{\sqrt{6}}$

$proj\overrightarrow{Q}\overrightarrow{P}=\frac{\sqrt{3}}{\sqrt{2}}$