CBSE Class 12 Mathematics — Vector Algebra

The value of p for which the vectors $\hat{i} + 2\hat{j} + 3\hat{k}$ and $2\hat{i} - p\hat{j} + \hat{k}$ are perpendicular to each other is

The value of m for which the points with position vectors $-\hat{i} - \hat{j} + 2\hat{k}$, $2\hat{i} + m\hat{j} + 5\hat{k}$ and $3\hat{i} + 11\hat{j} + 6\hat{k}$ are collinear, is

If $|\vec{a}| = 8$, $|\vec{b}| = 3$ and $|\vec{a} \times \vec{b}| = 12$, then the value of $|\vec{a} \cdot \vec{b}|$ is

Find the vector of magnitude 14 in the direction of vector QP, where P and Q are the points $(1, 3, 2)$ and $(-1, 0, 8)$ respectively.

Vectors $a = 3i - 2j + 2k$ and $b = i + 2k$ represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.