Let $\bar{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$,where $a_1, a_2, a_3$ and $|\bar{a}|$ are rational numbers. If $\bar{a}$ makes an angle of $45^{\circ}$ with $\bar{b} = \sqrt{2} \hat{i} + 3 \sqrt{2} \hat{j} + 4 \hat{k}$,then $\bar{a}$ lies in:

The angle between the vectors $2 \hat{k} - 3 \hat{j}$ and $\hat{i} - 2 \hat{k}$ is

If $p = i - 2j + 3k$ and $q = 3i + j + 2k,$ then a vector along $r$ which is a linear combination of $p$ and $q$ and also perpendicular to $q$ is

Two adjacent sides of a parallelogram are represented by $\vec{AB} = 2\hat{i} + 10\hat{j} + 11\hat{k}$ and $\vec{AD} = -\hat{i} + 2\hat{j} + 2\hat{k}$. The side $\vec{AD}$ is rotated by an acute angle $\theta$ in the plane of the parallelogram such that $\vec{AD}$ becomes $\vec{AD'}$. If $\vec{AD'}$ is perpendicular to $\vec{AB}$,find $\cos \theta$.

Let $a = 2\hat{i} - \hat{j} + 2\hat{k}$ and $b = 3\hat{i} - 2\hat{j} - 5\hat{k}$ be two vectors. Then the projection vector of $b$ on a vector perpendicular to $a$ is

The angle between two consecutive sides $\vec{a}$ and $\vec{b}$ of a parallelogram is $\frac{\pi}{6}$. Given $\vec{a} = (2, -2, 1)$ and $\vec{b} = 2|\vec{a}|$,the area of the parallelogram is . . . . . . sq. units.