If $\vec{A} = 2\hat{i} + 3\hat{j} + 8\hat{k}$ is perpendicular to $\vec{B} = -4\hat{i} + 4\hat{j} + \alpha\hat{k}$,then the value of $\alpha$ is:

The angle between force $\vec{F}=3 \hat{i}+4 \hat{j}-5 \hat{k}$ and displacement $\vec{d}=5 \hat{i}+4 \hat{j}+3 \hat{k}$ is

Show that $\vec{a} \cdot(\vec{b} \times \vec{c})$ is equal in magnitude to the volume of the parallelepiped formed by the three vectors $\vec{a}, \vec{b}$ and $\vec{c}$.

The area of the parallelogram having diagonals $\vec{d_1} = 3\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{d_2} = \hat{i} - 3\hat{j} + 4\hat{k}$ is:

If the vector $2\hat{i} + 3\hat{j} - \hat{k}$ is perpendicular to the vector $-4\hat{i} - 6\hat{j} + \lambda\hat{k}$,find the value of $\lambda$.

If $|\hat{a} \cdot \hat{b}| = \frac{1}{2}$,then $|\hat{a} - \hat{b}|$ may be $:-$