If vectors $\vec{A} = (2, -3, 1)$ and $\vec{B} = (3, 4, n)$ are perpendicular to each other,find the value of $n$.

Find the angle between force $F = (3 \hat{i} + 4 \hat{j} - 5 \hat{k})$ units and displacement $d = (5 \hat{i} + 4 \hat{j} + 3 \hat{k})$ units. Also,find the projection of $F$ on $d$.

Obtain the scalar product of two vectors in terms of their Cartesian components.

If diagonals of a parallelogram are $(5\hat i - 4\hat j + 3\hat k)$ and $(3\hat i + 2\hat j - \hat k)$,then its area is

$\overrightarrow A = 2\hat i + 4\hat j + 4\hat k$ and $\overrightarrow B = 4\hat i + 2\hat j - 4\hat k$ are two vectors. The angle between them will be ........ $^o$.

For three vectors $\vec{A} = (-x \hat{i} - 6 \hat{j} - 2 \hat{k})$,$\vec{B} = (-\hat{i} + 4 \hat{j} + 3 \hat{k})$ and $\vec{C} = (-8 \hat{i} - \hat{j} + 3 \hat{k})$,if $\vec{A} \cdot (\vec{B} \times \vec{C}) = 0$,then the value of $x$ is: