Let the volume of the tetrahedron with vertices $\hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}},-2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}},-\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+a \hat{\mathbf{k}}$ be $\frac{20}{3}$. Then the integral value of $a$ is

If $3 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, 7 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}, \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $-7 \hat{\mathbf{i}}-17 \hat{\mathbf{j}}+16 \hat{\mathbf{k}}$ are position vectors of the points $A, B, C$ and $D$ respectively, then the angle between $\mathbf{A B}$ and $\mathbf{C D}$ is

If $A(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}), B(\lambda \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}), C(-4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ and $D(-\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})$ are four points in space such that $\mathbf{A B}=x \mathbf{A C}+y \mathbf{A D}$ for some real number $x \neq 0, y \neq 0$, then $17(\lambda+9)=$

Let $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{b}$ be two vectors such that $\mathbf{a} \cdot \mathbf{b}=1$, $\cos (\mathbf{a} \cdot \mathbf{b})=\frac{1}{3}$ and the components of $\mathbf{b}$ w.r.t. $(\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}})$ be integers. Then, the number of possible vectors that represent $\mathbf{b}$ is