Question

If the scalar projection of the vector $xi-j+k$ on the vector $2i-j+5k$ is $\frac{1}{\sqrt{30}}$ then value of $x$ is equal to

• A. $\frac{-5}{2}$
• B. $6$
• C. $-6$
• D. $3$

Answer 1

The correct option is A $\frac{-5}{2}$

Scalar projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ $=\frac{|\overrightarrow{a}.\overrightarrow{b}|}{\left|\overrightarrow{b}\right|}$

let $\overrightarrow{a}=xi-j+k$ and $\overrightarrow{b}=2i-j+5k$

Then projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ $=\frac{(xi-j+k).(2i-j+5k)}{|2i-j+5k|}$

Given scalar projection $=\frac{1}{\sqrt{30}}$

$\Rightarrow \frac{1}{\sqrt{30}}=\frac{2x+1+5}{\sqrt{2^2+{(-1)}^2+5^2}}\Rightarrow \frac{1}{\sqrt{30}}=\frac{2x+1+5}{\sqrt{30}}\Rightarrow 2x+1+5=1\Rightarrow x=-\frac{5}{2}$