If the projection of the vector $\hat{i}+2 \hat{j}+\hat{k}$ on the sum of the two vectors $2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $-\lambda \hat{i}+2 \hat{j}+3 \hat{k}$ is $1,$ then $\lambda$ is equal to ..... .

If the angles between the sides of the triangle $ABC$ formed by $A(2,3,5)$,$B(-1,3,2)$ and $C(3,5,-2)$ are $\alpha, \beta$ and $\gamma$,then $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = $

In triangle $ABC$,the point $P$ divides $BC$ internally in the ratio $3:4$ and $Q$ divides $CA$ internally in the ratio $5:3$. If $AP$ and $BQ$ intersect in a point $G$,then $G$ divides $AP$ internally in the ratio

If $\theta$ is the angle between the vectors $\vec{a}$ and $\vec{b}$,then $\frac{|\vec{a} \times \vec{b}|}{|\vec{a} \cdot \vec{b}|}$ is equal to:

The points $A(5, -1, 1)$,$B(7, -4, 7)$,$C(1, -6, 10)$,and $D(-1, -3, 4)$ are vertices of a:

Let $\bar{a}$ and $\bar{b}$ be two vectors such that $|\bar{a}|=|\bar{b}|$ and $|\bar{a}+2 \bar{b}|=|2 \bar{a}-\bar{b}|$. If $\bar{c}$ is a vector parallel to $\bar{a}$,then the angle between $\bar{b}$ and $\bar{c}$ is (in $^{\circ}$)