Given

$\overrightarrow{a}=5\hat{i}-\hat{j}-3\hat{k}$

So, $\left|\overrightarrow{a}\right|=\sqrt{5^2+1^2+3^2}=\sqrt{35}$

$\overrightarrow{b}=\hat{i}+3\hat{j}+5\hat{k}$

So, $\sqrt{5^2+1^2+3^2}=\sqrt{35}$

So, $\overrightarrow{a}·\overrightarrow{b}=5-3-15=-13$

Now finding the projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ using the formula we get,

$\frac{\overrightarrow{a}·\overrightarrow{b}}{\left|\overrightarrow{b}\right|}=-\frac{13}{\sqrt{35}}$

And projection of $\overrightarrow{b}$ on $\overrightarrow{a}$ is given by,

$\frac{\overrightarrow{a}·\overrightarrow{b}}{\left|\overrightarrow{a}\right|}=-\frac{13}{\sqrt{35}}$

Note- The original answer key was dropped by NTA in Jee mains 1st feb 2023, here we have modified the options.