Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 1: Nagaland Board-2019

If $p\hat{i}+3\hat{j}$ is a vector of magnitude $5$, then the value of $p$ is

Find the value of $\lambda$ for which $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular if $\overrightarrow{a}=7\hat{i}-\lambda \hat{j}-7\hat{k}$ and $\overrightarrow{b}=4\hat{i}+5\hat{j}-\hat{k}$

If $\overrightarrow{a}=(\hat{i}-\hat{j}),\overrightarrow{b}=(3\hat{\mathit{j}}-\hat{\mathit{k}})\text{ and }\overrightarrow{c}=(7\hat{\mathit{i}}-\hat{\mathit{k}})$, find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$ and for which $\overrightarrow{c}·\overrightarrow{d}=4$.

Show that the points $\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}$ with position vectors $4\hat{i}+8\hat{j}+12\hat{k}$,$2\hat{i}+4\hat{j}+6\hat{k}, 3\hat{i}+5\hat{j}+4\hat{k}$ and$5\hat{i}+8\hat{j}+5\hat{k}$   respectively are coplanar.

The position vector of the point which divides the join of the points $2\overrightarrow{a}-3\overrightarrow{b}$ and $\overrightarrow{a}+\overrightarrow{b}$ in the ratio of $3: 1$ internally is

If the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ be such that $\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=\frac{\sqrt{2}}{3}$ and $\overrightarrow{a}\times \overrightarrow{b}$ is a unit vector, then the angle between $\overrightarrow{a} \text{and} \overrightarrow{b}$ is

Find $\left|\overrightarrow{a}\times \overrightarrow{b}\right|$, if $\overrightarrow{a}=\hat{i}-7\hat{j}+7\hat{k}$ and $\overrightarrow{b}=3\hat{i}-2\hat{j}+2\hat{k}$.

By using vectors, find the area of the triangle with vertices $A(1,1,2),B(2,3,5)$ and $C(1,5,5)$.