For non-coplanar vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, if the point of intersection of the line $\mathbf{r}=\mathbf{a}+t(\mathbf{b}-\mathbf{c})$ and the plane $\mathbf{r}=\mathbf{b}+\mathbf{c}+x(\mathbf{a}-\mathbf{b})+y(\mathbf{c}+\mathbf{a})$ is $l \mathbf{a}+m \mathbf{b}+n \mathbf{c}$, then $3 l+4 m+2 n=$

If the orthocentre of the triangle whose vertices are $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}, 5 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is $x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$, then

If the vectors $\mathbf{A B}=p \hat{\mathbf{i}}+q \hat{\mathbf{j}}+r \hat{\mathbf{k}}, \mathbf{A C}=s \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$, $\mathbf{C B}=3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ from $\triangle A B C$, then the values of $p, q, r$ and $s$ such that the area of that $\triangle A B C$ is $5 \sqrt{6}$ are

Let $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be three unit vectors such that $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\frac{1}{\sqrt{2}}(\mathbf{b}+\mathbf{c})$ and $\mathbf{b}$ is not parallel to $\mathbf{c}$. If $\alpha$ and $\beta$ are the angles between $\mathbf{a}, \mathbf{b}$ and $\mathbf{a}, \mathbf{c}$ respectively then $\alpha-\beta=$