Introduction to Scalar Triple Product

Write the value of $\left[\hat{i}+\hat{j} \hat{j}+\hat{k} \hat{k}+\hat{i}\right]$

Write the value of $\left[2\hat{i} 3\hat{j} 4\hat{k}\right]$.

Let $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}, \overrightarrow{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ and $\overrightarrow{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$ be three non-zero vectors such that $\overrightarrow{c}$ is a unit vector perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$. If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{6}$, then ${\left|\begin{array}{lll}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|}^2$ is equal to

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are non-coplanar vectors, then $\frac{\overrightarrow{a}·(\overrightarrow{b}\times \overrightarrow{c})}{(\overrightarrow{c}\times \overrightarrow{a})·\overrightarrow{b}}+\frac{\overrightarrow{b}·(\overrightarrow{a}\times \overrightarrow{c})}{\overrightarrow{c}·(\overrightarrow{a}\times \overrightarrow{b})}$ is equal to

If $\overrightarrow{a}$ lies in the plane of vectors $\overrightarrow{b}$ and $\overrightarrow{c}$, then which of the following is correct?

Evaluate the following: $\left[\hat{i}\hat{j}\hat{k}\right]+[ \hat{j}\hat{k}\hat{i}]+\left[\hat{k}\hat{i}\hat{j}\right]$

Evaluate the following: $\left[2\hat{i}\hat{j}\hat{k}\right]+\left[\hat{i}\hat{k}\hat{j}\right]+\left[\hat{k} \hat{j}2\hat{i}\right]$