Cross product practice · Medium

Let a unit vector $\widehat{O P}$ makes angles $\alpha, \beta, \gamma$ with the positive directions of the co-ordinate axes $O X, O Y$, $O Z$ respectively, where $\beta \in\left(0, \frac{\pi}{2}\right)$. If $\widehat{O P}$ is perpendicular to the plane through points $(1,2,3)$, $(2,3,4)$ and $(1,5,7)$, then which one of the following is true?

The vector $\vec{p}$ perpendicular to the vectors $\vec{a}=2 \hat{\imath}+3 \hat{\jmath}-\hat{k}$ and $\vec{b}=\hat{\imath}-2 \hat{\jmath}+3 \hat{k}$ and satisfying the condition $\vec{p} \cdot(2 \hat{\imath}-\hat{\jmath}+\hat{k})=-6$ is:

The area of triangle whose vertices are $(1,2,3),(2,5,-1)$ and $(-1,1,2)$ is

$\vec{a}=9 \hat{i}+2 \hat{k}, \vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}=7 \hat{i}-3 \hat{j}+2 \hat{k}$ are three given vectors. Let there be a $\vec{r}$ such that $(\vec{r} \times \vec{b})+(\vec{b} \times \vec{c})=\overrightarrow{0}$ and $\bar{r} \cdot \bar{a}=0$ then $\bar{r} \cdot \bar{c}$ is

The components of a vector a along and perpendicular to the non-zero vector $\mathbf{b}$ are respectively

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors equally inclined to each other at an angle $\alpha(\neq 0)$, then the angle between $\vec{a} \times \vec{b}$ and the plane containing $\vec{b}$ and $\vec{c}$ is

For any two vectors ${u}$ and ${v},(1-{u} \cdot {v})^2+\mid {u}+{v}+({u}$ $\times {v})\left.\right|^2$ is equal to

A tetrahedron has vertices $O(0,0,0), A(1,2,1), B(2,1,3)$ and $C(-1,1,2)$. Then the angle between the faces $O A B$ and $A B C$ will be

If the non-zero vectors ${\vec a}$ and ${\vec b}$ are perpendicular to each other, then the solution of the equation, $\vec r \times \vec a = \vec b$ is given by

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}-2 \hat{j}+\hat{k}$, then the vector $\vec{c}$ such that $\vec{a} \cdot \vec{c}=2$ and $\vec{a} \times \vec{c}=\vec{b}$ is

If $|\vec{a}|=1,|\vec{b}|=2, \vec{a} \cdot \vec{b}=4, \vec{c}=2(\vec{a} \times \vec{b})-3 \vec{b}$ then $\vec{b} \cdot \vec{c}$ equals