Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 1: JEE Main - 1 February 2023 Shift 2

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vector such that $\left|\overrightarrow{a}\right|=\sqrt{14}, \left|\overrightarrow{b}\right|=\sqrt{6}$ and $\left|\overrightarrow{a}\times \overrightarrow{b}\right|=\sqrt{48}$. Then ${\left(\overrightarrow{a}·\overrightarrow{b}\right)}^2$ is equal to _____ .

The foot of perpendicular from the origin $O$ to a plane $P$ which meets the co-ordinate axes at the point $A$, $B,C$ is $(2,a,4),a\in N$. If the volume of the tetrahedron $OABC$ is $144 {\mathrm{unit}}^3$, then which of the following points is NOT on $P$?

Let $\overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{c}=5\hat{i}-3\hat{j}+3\hat{k}$, be there(three) vector. If $\overrightarrow{r}$ is a vector such that, $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{b}$ and $\overrightarrow{r}·\overrightarrow{a}=0$, then $25{\left|\overrightarrow{r}\right|}^2$ is equal to

Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be three vectors such that $\left|\overrightarrow{a}\right|=\sqrt{31},4\left|\overrightarrow{b}\right|=\left|\overrightarrow{c}\right|=2$ and $2\left(\overrightarrow{a}\times \overrightarrow{b}\right)=3\left(\overrightarrow{c}\times \overrightarrow{a}\right)$. If the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{2\pi }{3}$, then ${\left(\frac{\overrightarrow{a}\times \overrightarrow{c}}{\overrightarrow{a}·\overrightarrow{b}}\right)}^2$ is equal to _____ .

$A\left(2,6,2\right),B\left(-4,0,\lambda \right),C\left(2,3,-1\right)$ and $D\left(4,5,0\right), \left|\lambda \right|\le 5$ are the vertices of a quadrilateral $ABCD$. If its area is $18$ square units, then $5-6\lambda$ is equal to _____ .

Let $\overrightarrow{v}=\alpha \hat{i}+2\hat{j}-3\hat{k}, \overrightarrow{w}=2\alpha \hat{i}+\hat{j}-\hat{k}$, and $\overrightarrow{u}$ be a vector such that $\left|\overrightarrow{u}\right|=\alpha >0$. If the minimum value of the scalar triple product $\left[\overrightarrow{u} \overrightarrow{v} \overrightarrow{w}\right]$ is $-\alpha \sqrt{3401}$, and ${\left|\overrightarrow{u}.\hat{i}\right|}^2=\frac{m}{n}$ where $m$ and $n$ are coprime natural numbers, then $m+n$ is equal to _____ .

Let $\overrightarrow{\mathrm{a}}=2\hat{\mathrm{i}}-7\hat{\mathrm{j}}+5\hat{\mathrm{k}} , \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}}$  be three given vectors. If $\overrightarrow{\mathrm{r}}$ is a vector such that $\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{r}}·\overrightarrow{\mathrm{b}}=0$, then $\left|\overrightarrow{\mathrm{r}}\right|$ is equal to:

Let $\overrightarrow{a}=5\hat{i}-\hat{j}-3\hat{k}$ and $\overrightarrow{b}=\hat{i}+3\hat{j}+5\hat{k}$ be two vectors. Then which one of the following statements is TRUE?