If $|\vec{a}|=5, |\vec{b}|=3, |\vec{c}|=4$ and $\vec{a}$ is perpendicular to both $\vec{b}$ and $\vec{c}$ such that the angle between $\vec{b}$ and $\vec{c}$ is $\frac{5 \pi}{6}$,then $[\vec{a} \vec{b} \vec{c}]=$

Given vectors $a, b, c$ such that $a \cdot (b \times c) = \lambda \neq 0$,the value of $\frac{(b \times c) \cdot (a + b + c)}{\lambda}$ is

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=\hat{i}-2\hat{j}+\hat{k}$,$\vec{c}=\hat{i}+3\hat{j}-2\hat{k}$,and $\vec{d}=2\hat{i}+\hat{j}-\hat{k}$ be four vectors. Let $l=\vec{b} \cdot \vec{c}$ and $m=\vec{b} \cdot \vec{a}$. Find the value of the scalar triple product $[(m\vec{b}+l\vec{a}) \quad \vec{b} \quad \vec{d}]$.

If $x, y$ and $z$ are non-zero real numbers and $\vec{a}=x \hat{i}+2 \hat{j}, \vec{b}=y \hat{j}+3 \hat{k}$ and $\vec{c}=x \hat{i}+y \hat{j}+z \hat{k}$ are such that $\vec{a} \times \vec{b}=z \hat{i}-3 \hat{j}+xy \hat{k}$ is not given,but $\vec{a} \times \vec{b}=6 \hat{i}-3 \hat{j}+\hat{k}$ is given as $z \hat{i}-3 \hat{j}+\hat{k}$,then the scalar triple product $[\vec{a} \vec{b} \vec{c}]$ is equal to:

$a$ is perpendicular to both $b$ and $c$. The angle between $b$ and $c$ is $\frac{2 \pi}{3}$. If $|a|=2$, $|b|=3$, and $|c|=4$, then $c \cdot (a \times b)$ is equal to (in $\sqrt{3}$)

If $a, b, c$ are any three coplanar unit vectors,then