Amit M Agarwal Solutions for Chapter: Product of Vectors, Exercise 2: Exercise for Session 2

If $\overrightarrow{a}, \overrightarrow{b} \text{and} \overrightarrow{c}$ are three non-zero vectors, then which of the following statement(s) is/are true?

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ are two given perpendicular vectors, which are non-zero. A vector $\overrightarrow{r}$ satisfying the equation $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{a},$ can be

If $\overrightarrow{a}$ and$\overrightarrow{b}$ are any two vectors, then possible integers(s) in the range of $\frac{3|\overrightarrow{a}+\overrightarrow{b}|}{2}+2|\overrightarrow{a}-\overrightarrow{b}|$ is

If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$ and $p$ is non-zero scalar such  that $p\overrightarrow{r}+\left(\overrightarrow{r}·\overrightarrow{b}\right)\overrightarrow{a}=\overrightarrow{c}$, then $\overrightarrow{r}$

In a four-dimensional space where unit vectors along axes are $\hat{i}, \hat{j}, \hat{k}, \hat{l}$ and ${\overrightarrow{a}}_{\mathrm{1}}, {\overrightarrow{a}}_{\mathrm{2}}, {\overrightarrow{a}}_{\mathrm{3}}, {\overrightarrow{a}}_{\mathrm{4}}$ are four non-zero vectors such that no vector can be expressed as a linear combination of others and $\left(\lambda –1\right)\left({\overrightarrow{a}}_{\mathrm{1}}–{\overrightarrow{a}}_{\mathrm{2}}\right)+\mu \left({\overrightarrow{a}}_{\mathrm{2}}+{\overrightarrow{a}}_{\mathrm{3}}\right)+\gamma \left({\overrightarrow{a}}_{\mathrm{3}}+ {\overrightarrow{a}}_{\mathrm{4}}–2{\overrightarrow{a}}_{\mathrm{2}}\right)+{\overrightarrow{a}}_{\mathrm{3}}+ \delta {\overrightarrow{a}}_{\mathrm{4}}=0$, then 

A vector $\hat{d}$ is equally inclined to three vectors $\overrightarrow{a} =\hat{i}–\hat{j}\mathrm{+}\hat{k}, \overrightarrow{b}= 2\hat{i}\mathrm{+}\hat{\mathrm{j}}$ and $\overrightarrow{c} = 3\hat{j}–2\hat{k}$.  Let $\overrightarrow{x}, \overrightarrow{y}, \overrightarrow{z}$ be  three vectors in the plane of$\overrightarrow{a}, \overrightarrow{b} ; \overrightarrow{b}, \overrightarrow{c} ; \overrightarrow{c}, \overrightarrow{a}$  respectively, then

If  $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are non-zero, non-collinear vectors such that a vector $\overrightarrow{p}\mathrm{=}\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \left(2\mathrm{\pi }–\left(\overrightarrow{a},\overrightarrow{b}\right)\right)\overrightarrow{c}$and $\overrightarrow{\mathit{q}}=\left|\overrightarrow{\mathit{a}}\right|\left|\overrightarrow{c}\right|\cos \left(\mathrm{\pi }– \left(\overrightarrow{\mathit{a}},\overrightarrow{\mathit{c}}\right)\right)\overrightarrow{\mathit{b}}$, then $\overrightarrow{p}+\overrightarrow{q}$ is

Given three vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ such that they are non-zero, non-coplanar vectors, then which of  the following are coplanar.