If $2\vec{a} + 3\vec{b} + \vec{c} = \vec{0}$,then $\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}$ is equal to

Let $\vec{a}=2 \hat{i}-3 \hat{j}-5 \hat{k}$ and $\vec{b}=3 \hat{i}+2 \hat{j}-5 \hat{k}$ be two vectors and $\vec{r}$ be a vector in the plane of $\vec{a}$ and $\vec{b}$. If $\vec{r}$ is orthogonal to the vector $5 \hat{i}-2 \hat{j}+3 \hat{k}$ and the magnitude of $\vec{r}$ is $\sqrt{94}$,then $|\vec{r} \cdot \vec{b}|=$

If $a = 2i + 2j - k$ and $b = 6i - 3j + 2k$,then what is the value of $a \times b$?

The area of the triangle with vertices $(1,2,0)$,$(1,0,2)$ and $(0,3,1)$ is.

If $\vec{a}_1$ is the component of vector $\vec{a}$ along the direction of vector $\vec{b}$,and $\vec{a}_2$ is the component of $\vec{a}$ perpendicular to $\vec{b}$,then $\vec{a}_1 \times \vec{a}_2 = \dots$

Given $|\vec{a}|=\sqrt{3}$,$|\vec{b}|=5$,$\vec{b} \cdot \vec{c}=10$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$. If $\vec{a}$ is perpendicular to $\vec{b} \times \vec{c}$,then the value of $|\vec{a} \times(\vec{b} \times \vec{c})|$ is