If $a = 4i + 6j$ and $b = 3j + 4k$,then the component of $a$ along $b$ is

If $\vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$ are orthogonal,then the value of $\lambda$ is:

If $a = 2i + j + 2k$ and $b = 5i - 3j + k$,then the projection of $b$ on $a$ is

If the moduli of the vectors $a, b, c$ are $3, 4, 5$ respectively and $a$ and $b + c$,$b$ and $c + a$,$c$ and $a + b$ are mutually perpendicular,then the modulus of $a + b + c$ is

The vector $\overline{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}$ lies in the plane of the vectors $\bar{b}=\hat{i}+\hat{j}$ and $\bar{c}=\hat{j}+\hat{k}$ and bisects the angle between $\bar{b}$ and $\bar{c}$. Then which one of the following gives possible values of $\alpha$ and $\beta$?

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\sqrt{3} \vec{c}=\overrightarrow{0}$,then the angle between $\vec{a}$ and $\vec{b}$ is