Embibe Experts Solutions for Exercise 2: EXERCISE 11.2

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-collinear vectors, then the value of $x$ for which vectors $\overrightarrow{\alpha }=\left(x-2\right)\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{\beta }=\left(3+2x\right)\overrightarrow{a}-2\overrightarrow{b}$ are collinear, is given by

Let $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three non-zero vectors such that any two of them are non-collinear. If $\overrightarrow{a}+\overrightarrow{b}$ is collinear with $\overrightarrow{c}$ and $\overrightarrow{b}+\overrightarrow{c}$ is collinear with $\overrightarrow{a},$ then the value of $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$ equals

If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=4\hat{i}+3\hat{j}+4\hat{k}$ and $\overrightarrow{c}=\hat{i}+\alpha \hat{j}+\beta \hat{k}$ are linearly dependent vectors and $\left|\overrightarrow{c}\right|=\sqrt{3},$ then 

If $\hat{x}, \hat{y}, \hat{z}$ are mutually perpendicular unit vectors and $a\hat{x}+a\hat{y}+c\hat{z}, \hat{x}+\hat{z}$ and $c\hat{x}+c\hat{y}+b\hat{z}$ are coplanar, then

If $\left|\overrightarrow{a}\right|=1, \left|\overrightarrow{b}\right|=2, \left|\overrightarrow{c}\right|=3$ and $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0,$ then the value of $\overrightarrow{a}·\overrightarrow{b}+\overrightarrow{b}·\overrightarrow{c}+\overrightarrow{c}·\overrightarrow{a}$ equals

ABCD is parallelogram with $\overrightarrow{AC}=\widehat{i}-2\widehat{j}+\widehat{k}$ and $\overrightarrow{BD}=-\widehat{i}+2\widehat{j}-5\widehat{k}.$ Then area of this parallelogram is equal to

The unit vector perpendicular to the plane of vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+\hat{j},$ is given by

If $\text{'}\mathrm{\theta }\text{'}$ is the angle between the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b},$ then $\sin \frac{\mathrm{\theta }}{2}$ is equal to