Subject Experts Solutions for Chapter: Vector Algebra, Exercise 1: Practice Exercise

If $\overrightarrow{b}$ and $\overrightarrow{c}$ are the position vectors of the points $B$ and $C$ respectively. Then, the position vector of the point $D$ such that $\overrightarrow{BD}=4\overrightarrow{BC}$ is

If the position vector $\overrightarrow{a}$ of the point $(5, n)$ is such that $\left|\overrightarrow{a}\right|=13$, then the value(s) of $n$ can be

If $\left|\overrightarrow{a}\right|=2$ and $\left|\overrightarrow{b}\right|=3$, then ${\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2+{\left|\overrightarrow{a}·\overrightarrow{b}\right|}^2$ is equal to

If the magnitude of difference of two unit vectors is $\sqrt{3}$, then the magnitude of sum of the two vectors is

If the vectors $\left(\alpha \hat{i}+\alpha \hat{j}+\gamma \hat{k}\right), \left(\hat{i}+\hat{k}\right)$ and $\left(\gamma \hat{i}+\gamma \hat{j}+\beta \hat{k}\right)$  lie on a plane, where $\alpha , \beta$ and $\gamma$ are distinct non-negative numbers, then $\gamma$ is

Consider the following inequalities in respect of the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.

$\text{I. }\left|\overrightarrow{a}+\overrightarrow{b}\right|\le \left|\overrightarrow{a}\right|+\left|\overrightarrow{b}\right|$

$\text{II. }\left|\overrightarrow{a}-\overrightarrow{b}\right|\ge \left|\overrightarrow{a}\right|-\left|\overrightarrow{b}\right|$

Which of the above is / are correct?

The vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $d$ are such that $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{d}$ and $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{d}$. Which of the following is/are correct?

$\text{I.}\left(\overrightarrow{a}-\overrightarrow{d}\right)\times \left(\overrightarrow{b}-\overrightarrow{c}\right)=0$

$\text{II.}\left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \left(\overrightarrow{c}\times \overrightarrow{d}\right)=0$

Select the correct answer using the code given below.

What is a vector of unit length orthogonal to both the vectors $\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}+3\hat{j}-\hat{k}$ ?