Triple Products

Let $\overrightarrow{a}=a_1\widehat{i}+a_2\widehat{j}+a_3\widehat{k}, \overrightarrow{b}=b_1\widehat{i}+b_2\widehat{j}+b_3\widehat{k}$ and $\overrightarrow{c}=c_1\widehat{i}+c_2\widehat{j}+c_3\widehat{k}$ be three non-zero vectors such that $\left|\overrightarrow{c}\right|=1$, angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{4}$ and $\overrightarrow{c}$ is perpendicular to $\overrightarrow{a}$ and $\overrightarrow{b}$, then

${\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|}^2=\lambda \left(a_1^2+a_2^2+a_3^2\right)\left(b_1^2+b_2^2+b_3^2\right)$ where $\lambda$ is equal to

Let $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ be three vectors such that $\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=\frac{\left|\overrightarrow{c}\right|}{2}=1$. If $\overrightarrow{a}\times \left(\overrightarrow{a}\times \overrightarrow{c}\right)+\overrightarrow{b}=\overrightarrow{0}$, then the acute angle between $\overrightarrow{a}$ and $\overrightarrow{c}$ is

Let unit vectors $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ are coplanar and a unit vector $\overrightarrow{d}$ is perpendicular to them. If $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=\frac{1}{6}\hat{i}-\frac{1}{3}\hat{j}+\frac{1}{3}\hat{k}$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $30°,$ then $\overrightarrow{c}$ is/are

If $\hat{a}, \hat{b}, \hat{c}$ are three unit vectors, $\hat{b}$ and $\hat{c}$ are non-parallel, such that $\hat{a}\times \left(\hat{b}\times \hat{c}\right)=\frac{\hat{b}+\hat{c}}{2},$ then the angle between $\hat{a}$ and $\hat{b}$ is

If $\overrightarrow{a}=\widehat{i}+\widehat{j}+\widehat{k},\overrightarrow{b}=2\widehat{i}+\lambda +\widehat{k},\overrightarrow{c}=\widehat{i}-\widehat{j}+4\widehat{k}$ and $\overrightarrow{a}.\left(\overrightarrow{b}\times \overrightarrow{c}\right)=10,$ then $\lambda$ is equal to

If $\overrightarrow{d}= \overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}$ is a non-zero vector and $\left|\left(\overrightarrow{d}\cdot \overrightarrow{c}\right)\left(\overrightarrow{a}\times \overrightarrow{b}\right)+\left(\overrightarrow{d}\cdot \overrightarrow{a}\right)\left(\overrightarrow{b}\times \overrightarrow{c}\right)+\left(\overrightarrow{d}\cdot \overrightarrow{b}\right)\left(\overrightarrow{c}\times \overrightarrow{a}\right)\right|=0,$ then

$\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three non-coplanar vectors such that $\left[\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c} \overrightarrow{a}-\overrightarrow{b}+\overrightarrow{c} 2\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}\right]$$=k \left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]$, then $k$ is

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

The scalar $\overrightarrow{\text{A}}·\left[\left(\overrightarrow{\text{B}}+\overrightarrow{\text{C}}\right)\times \left(\overrightarrow{\text{A}}+\overrightarrow{\text{B}}+\overrightarrow{\text{C}}\right)\right]$ equals

Find unit vector perpendicular to $\widehat{\dot{\mathbf{i}}}-\widehat{\dot{\mathbf{j}}}$ and coplanar with $\widehat{\dot{\mathbf{i}}}+2\widehat{\dot{\mathbf{j}}}$ and $\widehat{\dot{\mathbf{i}}}+3\widehat{\dot{\mathbf{j}}}$.

If the volume of parallelepiped formed by the vectors $\widehat{i}+\lambda \widehat{j}+\widehat{k},\widehat{j}+\lambda \widehat{k}$ and $\lambda \widehat{i}+\widehat{k}$ is minimum, then $\lambda$ is equal to:

Let $\overrightarrow{\mathbf{u}}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\overrightarrow{\mathbf{v}}$ are unit vectors such that $\overrightarrow{\mathbf{w}}\times \overrightarrow{\mathbf{u}}=\overrightarrow{\mathbf{v}}$, then the value of $\left[ \overrightarrow{\mathbf{u}}\mathbf{ }\overrightarrow{\mathbf{v}}\mathbf{ }\overrightarrow{\mathbf{w}}\mathrm{ }\right]$ is

Which of the following is not equal to $w.\left(u\times v\right)?$

If $\widehat{\mathrm{i}}\times \left[\left(\overrightarrow{a}-\widehat{\mathrm{j}}\right)\times \widehat{\mathrm{i}}\right]+\widehat{\mathrm{j}}\times \left[\left(\overrightarrow{a}-\widehat{\mathrm{k}}\right)\times \widehat{\mathrm{j}}\right]+\widehat{\mathrm{k}}\times \left[\left(\overrightarrow{a}-\widehat{\mathrm{i}}\right)\times \widehat{\mathrm{k}}\right]=0$ and $\overrightarrow{a}=x\widehat{\mathrm{i}}+y\widehat{\mathrm{j}}+z\widehat{\mathrm{k}}$, then the value of $8\left(x^3-xy+zx\right)$ is equal to

If $\overrightarrow{r}=\frac{a}{d}\hat{i}+\frac{b}{d}\hat{j}+\frac{c}{d}\hat{k}$ is a unit vector perpendicular to $\overrightarrow{p}=\hat{i}+\hat{j}-\hat{k}$ and coplanar with $\overrightarrow{p}=\hat{i}+\hat{j}-\hat{k} \& \overrightarrow{q}=\hat{i}+\hat{j}+\hat{k}$, then the value of $\frac{a^2+b^2+c^2+d^2}{b^2}$ is equal to 

If $\overrightarrow{A}=3\widehat{i}+\widehat{k}$ and $\overrightarrow{B}=2\widehat{i}+3\widehat{j}-6\widehat{k}$, then the value of $\left[\overrightarrow{A}+\overrightarrow{B} \overrightarrow{A}\times \overrightarrow{B} \overrightarrow{A}-\overrightarrow{B}\right]$ is equal to 

If $a\in [0, \infty )$, then the volume of parallelopiped formed by $\hat{i}+a\hat{k}, \hat{j}+a\hat{i}$ and $\hat{i}+a\hat{j}+a\hat{k}$ will be minimum when $a$ is

If $\overrightarrow{A}$ and $\overrightarrow{B}$ are two non-zero non-collinear vectors, then $\left[\overrightarrow{A} \overrightarrow{B} \hat{i}\right] \hat{i}+\left[\overrightarrow{A} \overrightarrow{B} \hat{j}\right] \hat{j}+\left[\overrightarrow{A} \overrightarrow{B} \hat{k}\right] \hat{k}$ will be same as 

If $\hat{i}\times \left(\overrightarrow{A}\times \hat{i}\right)+\hat{j}\times \left(\overrightarrow{A}\times \hat{j}\right)+\hat{k}\times \left(\overrightarrow{A}\times \hat{k}\right)=\lambda \overrightarrow{A}$, then $\lambda$ is equal to