vectors Questions

If $\vec{A}\times \vec{B}=\vec{B}\times \vec{A}$, then the angle between $\vec{A}$ and $\vec{B}$ is:

A force $\vec{F}=(5\hat{i}+3\hat{j}+2\hat{k})\text{N}$ is applied over a particle which displaces it from its origin to the point $\vec{r}=(2\hat{i}-\hat{j})\text{m}$. The work done on the particle in joules is:

Let $\vec{a},\vec{b}$ and $\vec{c}$ be non-zero vectors such that $(\vec{a}\times \vec{b})\times \vec{c}=\frac{1}{3}|\vec{b}||\vec{c}|\vec{a}.$ If $\theta$ is the acute angle between the vectors $\vec{b}$ and $\vec{c},$ then $\sin \theta$ equals

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of these are collinear. If the vector $\vec{a}+2\vec{b}$ is collinear with $\vec{c}$ and $\vec{b}+3\vec{c}$ is collinear with $\vec{a}$ then $\vec{a}+2\vec{b}+6\vec{c}$ equals

A particle is acted upon by constant forces $4\hat{i}+\hat{j}-3\hat{k}$ and $3\hat{i}+\hat{j}-\hat{k}$ which displace it from a point $\hat{i}+2\hat{j}+3\hat{k}$ to the point $5\hat{i}+4\hat{j}+\hat{k}.$ The work done in standard units by the forces is given by

Let $\vec{u},\vec{v},\vec{w}$ be such that $|\vec{u}|=1,|\vec{v}|=2,|\vec{w}|=3.$ If the projection $\vec{v}$ along $\vec{u}$ is equal to that of $\vec{w}$ along $\vec{u}$ and $\vec{v},\vec{w}$ are perpendicular to each other then $|\vec{u}-\vec{v}+\vec{w}|$ equals

If $\vec{a},\vec{b},\vec{c}$ are non-coplanar vectors and $\lambda$ is a real number, then the vectors $\vec{a}+2\vec{b}+3\vec{c},\lambda \vec{b}+4\vec{c}$ and $(2\lambda -1)\vec{c}$ are non-coplanar for