Given $\vec{P} + \vec{Q} = \vec{P} - \vec{Q}$. Under what condition is this true?

State the important condition for the addition of vectors.

The vector $\overrightarrow{OA}$ where $O$ is the origin is given by $\overrightarrow{OA} = 2\hat{i} + 2\hat{j}$. If it is rotated by $45^{\circ}$ in the anticlockwise direction about $O$,what will be the new vector?

The component of vector $\vec{A} = 2\hat{i} + 3\hat{j}$ along the vector $\vec{B} = \hat{i} + \hat{j}$ is:

If the vector $\overrightarrow{A} = 2\hat{i} + 4\hat{j} - 5\hat{k}$,find its direction cosines.

The direction cosines of the vector $\hat{i} + \hat{j} + \sqrt{2}\hat{k}$ are: