Find `lambda,` when the projection of ` vec a=lambda hat i+ hat j+4 hat k\ on\ vec b=2 hat i+6 hat j+3 hat k\ i s\ 4\ ` units.

Explanation

To solve the problem of finding \( \lambda \) when the projection of \( \vec{a} = \lambda \hat{i} + \hat{j} + 4 \hat{k} \) on \( \vec{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \) is 4 units, we can follow these steps:

### Step 1: Write the formula for the projection of vector \( \vec{a} \) onto vector \( \vec{b} \).

The projection of vector \( \vec{a} \) onto vector \( \vec{b} \) is given by the formula:
\[ \text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} \]
...