Embibe Experts Solutions for Exercise 3: EXERCISE 11.3

If $\left|\begin{array}{lll}a& a^2& 1+a^3\\ b& b^2& 1+b^3\\ c& c^2& 1+c^3\end{array}\right|=0$ and the vectors $A\equiv \left(1,a,a^2\right)$ $B\equiv \left(1,b,b^2\right), C\equiv \left(1,c,c^2\right)$ are non-coplanar, then the value of $\left(\mathit{a}\mathit{b}\mathit{c}\right)$ is equal to

If the vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are coplanar, then the value of $\left|\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\\ \overrightarrow{a}·\overrightarrow{a}& \overrightarrow{a}·\overrightarrow{b}& \overrightarrow{a}·\overrightarrow{c}\\ \overrightarrow{b}·\overrightarrow{a}& \overrightarrow{b}·\overrightarrow{b}& \overrightarrow{b}·\overrightarrow{c}\end{array}\right|=$

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three non-coplanar, non-zero vectors, then value of $\left(\overrightarrow{a}·\overrightarrow{a}\right)\left(\overrightarrow{b}\times \overrightarrow{c}\right)+\left(\overrightarrow{a}·\overrightarrow{b}\right)\left(\overrightarrow{c}\times \overrightarrow{a}\right)+\left(\overrightarrow{a}·\overrightarrow{c}\right)\left(\overrightarrow{a}\times \overrightarrow{b}\right)$ is equal to

The vectors $\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right), \overrightarrow{b}\times \left(\overrightarrow{c}\times \overrightarrow{a}\right)$ and $\overrightarrow{c}\times \left(\overrightarrow{a}\times \overrightarrow{b}\right)$ are

Given three unit vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c};$ no two of which are collinear satisfying $\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)=\frac{1}{2}\overrightarrow{b}$. The angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

If $\overrightarrow{a}$ and $\overrightarrow{\mathit{b}}$ are unit vectors, then the vector $\left(\overrightarrow{a}+\overrightarrow{b}\right)\times \left(\overrightarrow{a}\times \overrightarrow{b}\right)$ is parallel to the vector

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three unit vectors such that $\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)=\frac{1}{2}\left(\overrightarrow{b}+\overrightarrow{c}\right)$. If the vectors $\overrightarrow{b}$ and $\overrightarrow{c}$ are non-parallel, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

If $\overrightarrow{a}, \overrightarrow{c}, \overrightarrow{d}$ are non-coplanar vectors, then $\overrightarrow{d}·\left\{\overrightarrow{a}\times \left[\overrightarrow{b}\times \left(\overrightarrow{c}\times \overrightarrow{d}\right)\right]\right\}$ is equal to