If the vector $2\hat{i} + 3\hat{j} + 8\hat{k}$ is perpendicular to the vector $4\hat{j} - 4\hat{i} + \alpha\hat{k}$,then what is the value of $\alpha$?

If a vector $2\hat i + 3\hat j + 8\hat k$ is perpendicular to the vector $4\hat j - 4\hat i + \alpha \hat k$,then the value of $\alpha$ is:

If $\vec A = 2\hat i + \hat j - \hat k$,$\vec B = \hat i + 2\hat j + 3\hat k$ and $\vec C = 6\hat i - 2\hat j - 6\hat k$,then the angle between $(\vec A + \vec B)$ and $\vec C$ will be ....... $^o$.

Show that the magnitude of a vector is equal to the square root of the scalar product of the vector with itself.

If $\vec{P} = (k, 2, 3)$ and $\vec{Q} = (0, 3, k)$ and $\vec{P} \perp \vec{Q}$,then what is the value of $k$?

Show that the area of the triangle formed by the vectors $\vec{a}$ and $\vec{b}$ is one-half of the magnitude of $\vec{a} \times \vec{b}$.