Vector Triple Product

If  $\overrightarrow{a}=\hat{\mathit{i}}+2\hat{j}-2\hat{k}, \overrightarrow{b}=2\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{c}=\hat{i}+3\hat{j}-\hat{k},$ then $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})$ is equal to

If $\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)=0,$ then

If $\overrightarrow{u}=\hat{i}\times \left(\overrightarrow{a}\times \hat{i}\right)+\hat{j}\times \left(\overrightarrow{a}\times \hat{j}\right)+\hat{k}\times \left(\overrightarrow{a}\times \hat{k}\right),$ then

$\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)$ is coplanar with

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

Let $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ be three non-zero vectors such that no two of them are collinear and  $(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c} = \frac{1}{3}\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|\overrightarrow{a}$. If $\theta$ is the angle between vectors $\overrightarrow{b}$ and $\overrightarrow{c}$, then the value of  $\sin \theta$ is

Let $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ be three unit vectors such that $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\frac{\sqrt{3}}{2}(\overrightarrow{b}+\overrightarrow{c}).$ If $\overrightarrow{b}$ is not parallel to $\overrightarrow{c}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

If $\left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \overrightarrow{c}=\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)$, where $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ are any three vectors such that $\overrightarrow{a}·\overrightarrow{b}\ne 0, \overrightarrow{b}·\overrightarrow{c} \ne 0,$ then $\overrightarrow{a}$ and $\overrightarrow{c}$ are

Let $\overrightarrow{a}=\hat{j}-\hat{k}$ and $\overrightarrow{c}=\hat{i}-\hat{j}-\hat{k}$. Then, the vector $\overrightarrow{b}$ satisfying $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{c}=0$ and $\overrightarrow{a}·\overrightarrow{b}=3,$ is

If $\overrightarrow{a}=\frac{1}{\sqrt{10}}\left(3\hat{i}+\hat{k}\right)$ and $\overrightarrow{b}=\frac{1}{7}\left(2\hat{i}+3\hat{j}-6\hat{k}\right)$, then the value of $\left(2\overrightarrow{a}-\overrightarrow{b}\right)·\left[\left(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\right)\times \left(\overrightarrow{a}+2\overrightarrow{b}\right)\right]$ is

If $\left[\overrightarrow{a}\times \overrightarrow{b} \overrightarrow{b}\times \overrightarrow{c} \overrightarrow{c}\times \overrightarrow{a}\right]=\lambda {\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]}^2,$ then $\lambda$ is equal to

If $\overrightarrow{a}=\stackrel{⏜}{i}+\stackrel{⏜}{j}+\stackrel{⏜}{k}, \overrightarrow{a}·\overrightarrow{b}=1$ and $\overrightarrow{a}\times \overrightarrow{b}=\stackrel{⏜}{j}-\stackrel{⏜}{k}$, then $\overrightarrow{b}$ is equal to

Let $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three vectors along the adjacent edges of a tetrahedron, if $\left|a\right|=\left|b\right|=\left|c\right|=2$ and $\overrightarrow{a}. \overrightarrow{b}=\overrightarrow{b}. \overrightarrow{c}=\overrightarrow{c}. \overrightarrow{a}=2$, then volume of tetrahedron is

Let the area of faces, $∆OAB={\lambda }_1,∆OAC={\lambda }_2,\Delta OBC={\lambda }_3,\Delta ABC={\lambda }_4$ and $h_1, h_2, h_3, h_4$be the perpendicular height from $O$ to face $∆$$ABC$, $A$ to the face $∆OBC$, $B$ to the face $∆OAC, C$ to the face $∆OAB$ respectively, then the value of $\frac{1}{3}{\lambda }_1{h}_4+\frac{1}{3}{\lambda }_2{h}_3+\frac{1}{3}{\lambda }_3{h}_2+\frac{1}{3}{\lambda }_4{h}_1$ is equal to

Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{b}=-\hat{i}+\hat{j}+\hat{k} , \overrightarrow{c}=\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{d}=\hat{i}+\hat{j}-\hat{k}$. Then, the line of intersection of planes one determined by $\overrightarrow{a},\overrightarrow{b}$ and others determined by $\overrightarrow{c}, \overrightarrow{d}$ is perpendicular to

If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are any three non-zero vectors, then the component of $\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)$ perpendicular to $\overrightarrow{b}$ is

If $\hat{a}$ is a unit vector and projection of $\overrightarrow{x}$ along $\hat{a}$ is $2$ units and $\left(\hat{a}\times \overrightarrow{x}\right)+\overrightarrow{b}=\overrightarrow{x}$, then $\overrightarrow{x}$ is equal to

Let $\hat{\mathrm{u}}$ and $\hat{\mathrm{v}}$  are unit vectors and $\overrightarrow{\mathrm{w}}$ is a vector such that $\hat{\mathrm{u}}\text{×}\hat{\mathrm{v}}+\hat{\mathrm{u}}=\overrightarrow{\mathrm{w}}$ and $\overrightarrow{\mathrm{w}}\times \hat{\mathrm{u}}=\hat{\mathrm{v}}$, then find the value of $[\hat{\mathrm{u}}\hat{\mathrm{v}}\hspace{0.33em}\overrightarrow{\mathrm{w}}].$

If  $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are non-zero, non-collinear vectors such that a vector $\overrightarrow{p}\mathrm{=}\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \left(2\mathrm{\pi }–\left(\overrightarrow{a},\overrightarrow{b}\right)\right)\overrightarrow{c}$and $\overrightarrow{\mathit{q}}=\left|\overrightarrow{\mathit{a}}\right|\left|\overrightarrow{c}\right|\cos \left(\mathrm{\pi }– \left(\overrightarrow{\mathit{a}},\overrightarrow{\mathit{c}}\right)\right)\overrightarrow{\mathit{b}}$, then $\overrightarrow{p}+\overrightarrow{q}$ is

If $\overrightarrow{a}, \overrightarrow{b} \text{and} \overrightarrow{c}$ are three non-zero vectors, then which of the following statement(s) is/are true?