The projection of the vector $5\hat i+\hat j-3k$ on the vector $\hat i + 2\hat j - 3\hat k$ is

$\frac{16}{\sqrt{14}}$

The correct answer is Option (1) → $\frac{16}{\sqrt{14}}$

Let $\vec{a} = 5\hat{i} + \hat{j} - 3\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - 3\hat{k}$.

The projection of $\vec{a}$ on $\vec{b}$ is given by:

$\text{Proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$

First, compute the dot product:

$\vec{a} \cdot \vec{b} = (5)(1) + (1)(2) + (-3)(-3) = 5 + 2 + 9 = 16$

Now, compute $|\vec{b}| = \sqrt{1^2 + 2^2 + (-3)^2} = \sqrt{1 + 4 + 9} = \sqrt{14}$

So, the projection = $\frac{16}{\sqrt{14}}$