If vector a = 2i − j − k and vector b = λi + j + 5k and a ⊥ b, then λ = ?

If $\vec{a} = 2\hat{i} - \hat{j} - \hat{k}$ and $\vec{b} = \lambda\hat{i} + \hat{j} + 5\hat{k}$ and $\vec{a} \perp \vec{b}$, then $\lambda = \_$

If two vectors are perpendicular, their dot product is zero ($\vec{a} \cdot \vec{b} = 0$).

Vector projection $\vec{a} = \hat{i}$ on $\vec{b} = \hat{j}$ is

The formula for the vector projection of $\vec{a}$ on $\vec{b}$ is $\text{proj}_{\vec{b}} \vec{a} = \left( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \right) \vec{b}$.

Given:
$\vec{a} = \hat{i} = 1\hat{i} + 0\hat{j} + 0\hat{k}$
$\vec{b} = \hat{j} = 0\hat{i} + 1\hat{j} + 0\hat{k}$

Scalar projection vector $\hat{i} - \hat{j} - \hat{k}$ on vector $3\hat{i} + \hat{j} + 3\hat{k}$ is

The formula for the scalar projection of $\vec{u}$ on $\vec{v}$ is $\text{comp}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|}$.

Given:
$\vec{u} = \hat{i} - \hat{j} - \hat{k}$
$\vec{v} = 3\hat{i} + \hat{j} + 3\hat{k}$