If the vectors $\overline{a}=\hat{i}-\hat{j}+2 \hat{k}$,$\overline{b}=2 \hat{i}+4 \hat{j}+\hat{k}$,and $\overline{c}=\lambda \hat{i}+\hat{j}+\mu \hat{k}$ are mutually orthogonal,then $(\lambda, \mu) = $

The projection of vector $\vec{a}$ along vector $\vec{b}$ is:

The values of $x$ for which the angle between the vectors $\bar{a} = 2x^2 \hat{i} + 4x \hat{j} + \hat{k}$ and $\bar{b} = 7 \hat{i} - 2 \hat{j} + x \hat{k}$ is obtuse,are

If $\vec{a} = 2\hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = 6\hat{i} - 3\hat{j} + 2\hat{k}$,then find $\vec{a} \cdot \vec{b}$.

If $a$ and $b$ are unit vectors and $\theta$ $(0 < \theta < \pi)$ is the angle between them, then the value of $|a + b| / |a - b|$ is equal to

If the position vectors of the vertices $A, B, C$ of a triangle $ABC$ are $4 \hat{\imath} + 7 \hat{\jmath} + 8 \hat{k}$,$2 \hat{\imath} + 3 \hat{\jmath} + 4 \hat{k}$,and $2 \hat{\imath} + 5 \hat{\jmath} + 7 \hat{k}$ respectively,then the position vector of the point where the bisector of angle $A$ meets $BC$ is