Let $A(2,3,-1), B(4,1,0), C(-1,-1,1)$ be the vertices of a $\triangle A B C$. Let $D$ be the point where the bisector of $B A C$ meet the side $B C$. Then, the direction ratios of $A D$ are

If a plane passing through the points $(2,3,0),(0,-5,2)$ and ( $-2,0,3$ ) meets the $X, Y$ and $Z$-axes in $A, B$ and $C$ respectively, then $A=$

The point which lies on the plane passing through the point $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, 3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}},-3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$ is

If the angle between the planes $\mathbf{r} \cdot(11 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\alpha \hat{\mathbf{k}})=7$ and $\mathbf{r} \cdot(2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})=5$ is $\frac{\pi}{2}$, then $\alpha=$