R D Sharma Solutions for Chapter: Scalar Triple Product, Exercise 2: VERY SHORT ANSWER TYPE QUESTIONS (VSAQs)

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

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

Find the volume of the parallelepiped with its edges represented by the vectors $\hat{i}+\hat{j}, \hat{i}+2\hat{j}$ and $\hat{i}+\hat{j}+\pi \hat{k}$ .

[Write answer excluding the units]

If the vectors $\left({\sec }^{2}A\right)\hat{i}+\hat{j}+\hat{k}, \hat{i}+\left({\sec }^{2}B\right)\hat{j}+\hat{k}, \hat{i}+\hat{j}+\left({\sec }^{2}C\right)\hat{k}$ are coplanar, then find the value of ${cosec}^{2}A+{cosec}^{2}B+{cosec}^{2}C$.

For any two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ of magnitudes $3$ and $4$ respectively, write the value of $\left[\begin{array}{lll}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{a}\times \overrightarrow{b}\end{array}\right]+(\overrightarrow{a}·\overrightarrow{b}{)}^2$.

If $[3\overrightarrow{a}+7\overrightarrow{b} \overrightarrow{c} \overrightarrow{d}]=\lambda [\overrightarrow{a} \overrightarrow{c} \overrightarrow{d}]+\mu [\overrightarrow{b}\overrightarrow{c} \overrightarrow{d}]$ then find the value of $\lambda +\mu$.

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$are non-coplanar vectors, then find the value of $\frac{\overrightarrow{a}·\left(\overrightarrow{b}\times \overrightarrow{c}\right)}{\left(\overrightarrow{c}\times \overrightarrow{a}\right)·\overrightarrow{b}}+\frac{\overrightarrow{b}·\left(\overrightarrow{a}\times \overrightarrow{c}\right)}{\overrightarrow{c}·\left(\overrightarrow{a}\times \overrightarrow{b}\right)}$.

Find $\overrightarrow{a}·(\overrightarrow{b}\times \overrightarrow{c})$, if $\overrightarrow{a}=2\hat{i}+\hat{j}+3\hat{k},\overrightarrow{b}=-\overrightarrow{i}+2\overrightarrow{j}+\overrightarrow{k}\text{ and }\overrightarrow{c}=3\hat{i}+\hat{j}+2\hat{k}$.