$OA$ and $OB$ are two vectors of magnitudes $5$ and $6$ respectively. If $\angle BOA = 60^{\circ}$,then $OA \cdot OB$ is equal to

The angle between the vectors $a + b$ and $a - b$,when $a = (1, 1, 4)$ and $b = (1, -1, 4)$ is .............. $^o$

If $\bar{a}=\hat{i}+2 \hat{j}+\hat{k}$,$\bar{b}=\hat{i}-\hat{j}+\hat{k}$,and $\bar{c}=\hat{i}+\hat{j}-\hat{k}$,then a vector in the plane of $\bar{a}$ and $\bar{b}$,whose projection on $\bar{c}$ is $\frac{1}{\sqrt{3}}$,is

If the vectors $6i - 2j + 3k$,$2i + 3j - 6k$,and $3i + 6j - 2k$ are the position vectors of the vertices of a triangle,then the triangle is

The equation of the perpendicular bisector of the line segment joining the points whose position vectors are $a$ and $b$ respectively is

If $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$,$\vec{b}=\hat{i}-\hat{j}+4\hat{k}$,and $\vec{c}=\hat{i}+\hat{j}+\hat{k}$ are such that $\vec{a}+\lambda\vec{b}$ is perpendicular to $\vec{c}$,then the value of $\lambda$ is