The projection of the vector $2\hat i-\hat j+3\hat k$ on the vector $3\hat i +2\hat j+6\hat k$ is

$\frac{22}{7}$

The correct answer is Option (1) → $\frac{22}{7}$

Given vectors:

$\vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}$

$\vec{b} = 3\hat{i} + 2\hat{j} + 6\hat{k}$

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

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

$\vec{a} \cdot \vec{b} = (2)(3) + (-1)(2) + (3)(6) = 6 - 2 + 18 = 22$

$|\vec{b}| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7$

Therefore, the projection of $\vec{a}$ on $\vec{b}$ is $\frac{22}{7}$.