Question

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

• A. $\frac{1}{\sqrt{6}}$
• B. $\sqrt{6}$
• C. $\frac{2}{\sqrt{6}}$
• D. $\frac{3}{\sqrt{6}}$

Hint:

Projection = $\frac{ \vec{a} \cdot \vec{b} }{ |\vec{b}| }$ when you need only the scalar value.

Answer 1

The Correct Option is A

The projection of $\vec{a}$ on $\vec{b}$ is: \[ \text{Proj}_{\vec{b}}\, \vec{a} = \frac{ \vec{a} \cdot \vec{b} }{ |\vec{b}| }. \] Here, \[ \vec{a} = \hat{i}, \vec{b} = \hat{i} + \hat{j} + 2\hat{k}. \] Dot product: \[ \vec{a} \cdot \vec{b} = (1)(1) + (0)(1) + (0)(2) = 1. |\vec{b}| = \sqrt{1^2 + 1^2 + 2^2} = \sqrt{6}. \] So, \[ \text{Projection} = \frac{1}{\sqrt{6}}. \]