If $\vec{P} = b \hat{i} + 6 \hat{j} + \hat{k}$ and $\vec{Q} = \hat{i} - a \hat{j} + 4 \hat{k}$ are perpendicular to each other,and $3b - a = 5$,find the values of $a$ and $b$.

Two adjacent sides of a parallelogram are represented by the two vectors $\hat{i} + 2\hat{j} + 3\hat{k}$ and $3\hat{i} - 2\hat{j} + \hat{k}$. What is the area of the parallelogram?

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

If a vector $\vec A$ is parallel to another vector $\vec B$,then the resultant of the vector $\vec A \times \vec B$ will be equal to

The vectors $\overrightarrow P = a\hat i + a\hat j + 3\hat k$ and $\overrightarrow Q = a\hat i - 2\hat j - \hat k$ are perpendicular to each other. The positive value of $a$ is

If $\overrightarrow{A} = (2\hat{i} + 3\hat{j} - \hat{k}) \; m$ and $\overrightarrow{B} = (\hat{i} + 2\hat{j} + 2\hat{k}) \; m$,the magnitude of the component of vector $\overrightarrow{A}$ along vector $\overrightarrow{B}$ will be $...... \; m$.