If $\bar{a}$ and $\bar{b}$ are unit vectors such that $(\bar{a} + 2\bar{b})$ and $(5\bar{a} - 4\bar{b})$ are perpendicular to each other,then the angle between $\bar{a}$ and $\bar{b}$ is .....$^o$.

If $3 \hat{j}$,$4 \hat{k}$ and $3 \hat{j}+4 \hat{k}$ are the position vectors of the vertices $A$,$B$,and $C$ respectively of $\triangle ABC$,then the position vector of the point in which the bisector of $\angle A$ meets $BC$ is

The vector $b = 3j + 4k$ is to be written as the sum of a vector $b_1$ parallel to $a = i + j$ and a vector $b_2$ perpendicular to $a$. Then $b_1 = $

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 $\cos \theta$.

$P$ is the circumcentre of $\triangle ABC$. If the position vectors of $A, B, C$ and $P$ are $\bar{a}, \bar{b}, \bar{c}$ and $\frac{\bar{a}+\bar{b}+\bar{c}}{4}$ respectively,then the position vector of the orthocentre of this triangle is

Let a unit vector which makes an angle of $60^{\circ}$ with $2 \hat{i}+2 \hat{j}-\hat{k}$ and an angle of $45^{\circ}$ with $\hat{i}-\hat{k}$ be $\overrightarrow{C}$. Then $\overrightarrow{C}+\left(-\frac{1}{2} \hat{i}+\frac{1}{3 \sqrt{2}} \hat{j}-\frac{\sqrt{2}}{3} \hat{k}\right)$ is :