Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 1: NDA & NA EE 2019 Paper 1 Mathematics

If a line has direction ratios $<a+b, b+c, c+a>,$ then what is the sum of the squares of its direction cosines?

If $\hat{a}$ is a unit vector in the $XY$-plane making an angle $30°$ with the positive $X$-axis, then what is $\hat{a}$ equal to?

Let $A$ be a point in space such that $\left|\mathrm{OA}\right|=12,$ where $O$ is the origin. If $OA$ is inclined at angles $45°$ and $60°$ with $X$-axis and $Y$-axis respectively, then what is $OA$ equal to?

Two adjacent sides of a parallelogram are $2\hat{\mathrm{i}}-4\hat{\mathrm{j}}+5\hat{\mathrm{k}}$ and $\hat{\mathrm{i}}-2\hat{\mathrm{j}}-3\widehat{\mathrm{k}.}$ What is the magnitude of dot product of vectors which represent its diagonals?

If $|\overrightarrow{a}\times \overrightarrow{b}{|}^2+|\overrightarrow{a}·\overrightarrow{b}{|}^2=144$ and $\left|\overrightarrow{a}\right|=4,$ then what is $\left|\overrightarrow{\mathrm{b}}\right|$equal to?

If the vectors $\overrightarrow{a}=2\hat{i}-3\hat{j}+\hat{k}, \overrightarrow{b}=\hat{i}+2\hat{j}-3\hat{k}$  and $\overrightarrow{c}=\hat{j}+p\hat{k}$ are coplanar, then what is the value of $p?$

Consider the following statements.

$1.$ The magnitude of $a\times b$ is same as the area of a triangle with sides $a$ and $b$.

$2.$ If $a\times b=0,$ where $a\ne 0, b\ne 0,$ then $a=\lambda b.$

Which of the above statements is/are correct?

If $a$ and $b$ are unit vectors and $\theta$ is the angle between them, then what is ${\sin }^2\left(\frac{\theta }{2}\right)$ equal to?