$(a \cdot i)i + (a \cdot j)j + (a \cdot k)k = $

Show that the vector $\hat{i}+\hat{j}+\hat{k}$ is equally inclined to the axes $OX, OY,$ and $OZ$.

Let a vector $\alpha \hat{i}+\beta \hat{j}$ be obtained by rotating the vector $\sqrt{3} \hat{i}+\hat{j}$ by an angle $45^{\circ}$ about the origin in the counterclockwise direction in the first quadrant. Then the area of the triangle having vertices $(\alpha, \beta), (0, \beta)$ and $(0,0)$ is equal to

$A$ vector of magnitude $\sqrt{2}$ units along the internal bisector of the angle between the vectors $\vec{a} = 2 \hat{i} - 2 \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2 \hat{j} + 2 \hat{k}$ is

If $ABCD$ is a parallelogram with $AC$ and $BD$ as diagonals,then which of the following is true regarding the vector relationship between the diagonals and sides?

$A, B, C, D$ are any four points. If $E$ and $F$ are midpoints of $AC$ and $BD$ respectively,then $\vec{AB} + \vec{CB} + \vec{CD} + \vec{AD} =$