Embibe Experts Solutions for Exercise 7: Assignment

The angles of a triangle, two of whose sides are represented by vectors $\sqrt{3}\left(\hat{a}\times \overrightarrow{b}\right)$ and $\overrightarrow{b}-\left(\hat{a}·\overrightarrow{b}\right)\overrightarrow{a}$ where $\overrightarrow{b}$ is a non-zero vector and $\hat{a}$ is a unit vector in the direction of $\overrightarrow{a},$ are

If $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ are non coplanar vectors and $x$ is a real number, then the vectors $\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c},$ $x\overrightarrow{b}+y\overrightarrow{c}$ and $\left(2x-1\right)\overrightarrow{c}$ are coplanar for

If $\overrightarrow{x}$ and $\overrightarrow{y}$ be two non-zero vectors such that $\left|\overrightarrow{x}+\overrightarrow{y}\right|=\left|\overrightarrow{x}-2\overrightarrow{y}\right|$, then

Let $A$ and $B$ be two points in space with position vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ respectively, then the real number $k$ such that the system of equations $\left|3\overrightarrow{r}-2\overrightarrow{a}-\overrightarrow{b}\right|=\left|\overrightarrow{a}-\overrightarrow{b}\right|$ and $\left\{\overrightarrow{r}-k\overrightarrow{a}-\left(1-k\right)\overrightarrow{b}\right\}·\left(\overrightarrow{a}-\overrightarrow{b}\right)=0$ does not have any solution, is

The vector(s) which is/are coplanar with vectors $\hat{i}+\hat{j}+2\hat{k}$ and $\hat{i}+2\hat{j}+\hat{k},$ and perpendicular to the vector $\hat{i}+\hat{j}+\hat{k}$ is/are

Let $L_1$ and $L_2$ denote the lines $\overrightarrow{r}=\hat{i}+\lambda \left(-\hat{i}+2\hat{j}+2\hat{k}\right), \lambda \in \mathrm{ℝ}$ and $\overrightarrow{r}=\mu \left(2\hat{i}-\hat{j}+2\hat{k}\right), \mu \in \mathrm{ℝ}$ respectively. If $L_3$ is a line which is perpendicular to both $L_1$and $L_2$ and cuts both of them, then which of the following options describe(s) $L_3?$

Consider the cube in the first octant with sides $OP, OQ$ and $OR$ of length $1,$ along the $x$-axis, $y$-axis and $z$-axis, respectively, where $O\left(0, 0, 0\right)$ is the origin. Let $S\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal OT. If $\overrightarrow{p}=\overrightarrow{SP}, \overrightarrow{q}=\overrightarrow{SQ}, \overrightarrow{r}=\overrightarrow{SR}$ and $\overrightarrow{t}=\overrightarrow{ST},$ then the value of $2|\left(\overrightarrow{p}\times \overrightarrow{q}\right)\times \left(\overrightarrow{r}\times \overrightarrow{t}\right)|$ is _____.

In a triangle $PQR,$ let $\overrightarrow{a}=\overrightarrow{QR}, \overrightarrow{b}=\overrightarrow{RP}$ and $\overrightarrow{c}=\overrightarrow{PQ}.$ If $\left|\overrightarrow{a}\right|=3, \left|\overrightarrow{b}\right|=4$ and $\frac{\overrightarrow{a}·(\overrightarrow{c}-\overrightarrow{b})}{\overrightarrow{c}·(\overrightarrow{a}-\overrightarrow{b})}=\frac{\left|\overrightarrow{a}\right|}{\left|\overrightarrow{a}\right|+\left|\overrightarrow{b}\right|}$

then the value of $|\overrightarrow{a}\times \overrightarrow{b}{|}^2$ is