If the vectors $ai + bj + ck$ and $\pi + qj + rk$ are perpendicular, then

Find the distance of the point $P(-\hat{i} + 2\hat{j} + 6\hat{k})$ from the line passing through the point $A(2\hat{i} + 3\hat{j} - 4\hat{k})$ and parallel to the vector $\vec{b} = 6\hat{i} + 3\hat{j} - 4\hat{k}$.

Let $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=2 \hat{j}-3 \hat{k}$. If $\vec{b}=\vec{c}-\vec{d}$, $\vec{a}$ is parallel to $\vec{c}$, and $\vec{a}$ is perpendicular to $\vec{d}$, then $\vec{c}+\vec{d}=$

The angle between two vectors $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + \hat{j} - \hat{k}$ is . . . . . . .

If $a, b$ and $c$ are unit vectors such that $a+b+c=0$,then the angle between $a$ and $b$ is

If $a = 2i + j + 2k$ and $b = 5i - 3j + k$,then the projection of $b$ on $a$ is