Scalar and Vector Triple Products

If $\overrightarrow{a}=5\hat{i}+6\hat{j}+7\hat{k}$, $\overrightarrow{b}=2\hat{i}+3\hat{j}+4\hat{k}$,$\overrightarrow{c}=\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{d}=3\hat{i}+\hat{j}+5\hat{k}$, then find $\left(\overrightarrow{a}\times \overrightarrow{b}\right)·\left(\overrightarrow{c}\times \overrightarrow{d}\right)$.

If $\overrightarrow{a}=3\hat{i}+4\hat{j}+5\hat{k}$, $\overrightarrow{b}=2\hat{i}+3\hat{j}+4\hat{k}$,$\overrightarrow{c}=\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{d}=3\hat{i}+\hat{j}+5\hat{k}$, then find $\left(\overrightarrow{a}\times \overrightarrow{b}\right)·\left(\overrightarrow{c}\times \overrightarrow{d}\right)$.

If $\overrightarrow{a}=3\hat{i}+4\hat{j}+5\hat{k}$, $\overrightarrow{b}=2\hat{i}+3\hat{j}+4\hat{k}$,$\overrightarrow{c}=\hat{i}+3\hat{j}+5\hat{k}$ and $\overrightarrow{d}=3\hat{i}+\hat{j}+5\hat{k}$, then find $\left(\overrightarrow{a}\times \overrightarrow{b}\right)·\left(\overrightarrow{c}\times \overrightarrow{d}\right)$.

If $\overrightarrow{a}=2\hat{i}+3\hat{j}+5\hat{k}$, $\overrightarrow{b}=4\hat{i}+3\hat{j}+3\hat{k}$,$\overrightarrow{c}=\hat{i}+2\hat{j}+5\hat{k}$ and $\overrightarrow{d}=2\hat{i}+\hat{j}+5\hat{k}$, then find $\left(\overrightarrow{a}\times \overrightarrow{b}\right)·\left(\overrightarrow{c}\times \overrightarrow{d}\right)$.

If $\overrightarrow{a}=2\hat{i}+3\hat{j}+4\hat{k}$, $\overrightarrow{b}=4\hat{i}+2\hat{j}+3\hat{k}$,$\overrightarrow{c}=\hat{i}+3\hat{j}+5\hat{k}$ and $\overrightarrow{d}=2\hat{i}+\hat{j}+3\hat{k}$, then find $\left(\overrightarrow{a}\times \overrightarrow{b}\right)·\left(\overrightarrow{c}\times \overrightarrow{d}\right)$.

If $\overrightarrow{a}=3\hat{i}-\hat{j}+4\hat{k}, \overrightarrow{b}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overrightarrow{c}=-5\hat{i}+2\hat{j}+3\hat{k},$ then $\overrightarrow{a}·(\overrightarrow{b}\times \overrightarrow{c})$ is

$\overrightarrow{a}=\hat{ı}+\hat{ȷ}+\hat{\mathit{k}}, \overrightarrow{b}=\hat{ı}-\hat{ȷ}+2\hat{\mathit{k}}$ and $\overrightarrow{c}=x\hat{ı}+\left(x-1\right)\hat{ȷ}-\hat{\mathit{k}}$. If the vector $\overrightarrow{c}$ lies in the plane of $\overrightarrow{a}$ and $\overrightarrow{b}$, then $x=$

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

Let $\overrightarrow{a}$ be a unit vector coplanar with $\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}$ and $2\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+\widehat{\mathrm{k}}$ such that $\overrightarrow{a}$ is perpendicular to $\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+\widehat{\mathrm{k}}$. If the projection of $\overrightarrow{a}$ along $\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}$ is $\lambda$ units, then the value of $\frac{1}{{\lambda }^2}$ is equal to

The points $A\left(4,\mathrm{ }5,\mathrm{ }1\right),B\left(0,-1,-1\right),C\left(3,\mathrm{ }9,\mathrm{ }4\right)$ and $D\left(-4,\mathrm{ }4,\mathrm{ }4\right)$ are

If $a\left(\stackrel{-}{\alpha }\times \stackrel{-}{\beta }\right)+b\left(\stackrel{-}{\beta }\times \stackrel{-}{\gamma }\right)+c\left(\stackrel{-}{\gamma }\times \stackrel{-}{a}\right)= \stackrel{-}{0}$ and atleast one of the scalars a, b, c is non-zero, then the vectors $\stackrel{-}{\alpha }, \stackrel{-}{\beta }, \stackrel{-}{\gamma }$ are

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 vectors $\overrightarrow{b} \& \overrightarrow{c}$ then $\sin \theta =$

If $\stackrel{-}{a}=2\stackrel{-}{i}-3\stackrel{-}{j}+5\stackrel{-}{k}, \stackrel{-}{b}=3\stackrel{-}{i}-4\stackrel{-}{j}+5\stackrel{-}{k} and \stackrel{-}{c}=5\stackrel{-}{i}-3\stackrel{-}{j}-2\stackrel{-}{k},$ then the volume of the parallelepiped with co - terminus edges $\stackrel{-}{a}+\stackrel{-}{b}, \stackrel{-}{b}+\stackrel{-}{c}, \stackrel{-}{c}+\stackrel{-}{a}$ is

If one of the vertices of a parallelepiped is origin and its edges are $\stackrel{-}{OA}, \stackrel{-}{OB}$ and $\stackrel{-}{OC}$ where A(4, 3, 1), B(3, 1, 2) and C(5, 2, 1), then find the volume of this parallelepiped.

For three vectors $\overrightarrow{u}, \overrightarrow{v}, \overrightarrow{w}$, which of the following expressions are meaningful

For three vectors $\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$ which of the following expressions is not equal to any of the remaining three

$\overrightarrow{a} . \left(\overrightarrow{a} \times \overrightarrow{b}\right)=$

$\hat{i}.\left(\hat{j}\times \hat{k}\right)+\hat{j}.\left(\hat{k}\times \hat{i}\right)+\hat{k}.\left(\hat{i}\times \hat{j}\right)=$

$\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{a} \times \overrightarrow{b}\right]$

is equal to

If $\overrightarrow{a}$ and $\overrightarrow{b}$ be parallel vectors, then $\left[\overrightarrow{a} \overrightarrow{c} \overrightarrow{b}\right]=$