Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 1: Exercise 1

If $\left|x_1\right|>\left|y_1\right|+\left|z_1\right|, \left|y_2\right|>\left|x_2\right|+\left|z_2\right|, \left|z_3\right|>\left|x_3\right|+\left|y_3\right|$, then $x_1\hat{i}+y_1\hat{j}+z_1{\hat{k}}_1, x_2\hat{i}+y_2\hat{j}+z_2\hat{k}$ and $x_3\hat{i}+y_3\hat{j}+z_3\hat{k}$ are

Let the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ be perpendicular to each other and the unit vector $\overrightarrow{c}$ be inclined at an angle $\theta$ to both $\overrightarrow{a}$ and $\overrightarrow{b}$. If $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+z\left(\overrightarrow{a}\times \overrightarrow{b}\right)$, then

A vector whose modulus is $\sqrt{51}$ and makes the same angle with $\overrightarrow{a}=\frac{\hat{i}-2\hat{j}+2\hat{k}}{3}, \overrightarrow{b}=\frac{-4\hat{i}-3\hat{k}}{5}$ and  $c=\hat{j},$ will be

Let $\overrightarrow{v}=2\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{w}=\hat{i}+3\hat{k}$. If $\overrightarrow{u}$ is a unit vector, then for the maximum value of the scalar triple product $\left[\overrightarrow{u} \overrightarrow{v} \overrightarrow{w}\right], \overrightarrow{u}=$

Which of the following statement$\left(\mathrm{s}\right)$ is/are correct?

$\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ and $\overrightarrow{d}$ are the position vectors of the points $A\equiv \left(x, y, z\right); B\equiv \left(y, -2z, 3x\right); C\equiv \left(2z, 3x, -y\right)$ and $D\equiv \left(1, -1, 2\right)$ respectively. If $\left|\overrightarrow{a}\right|=2\sqrt{3}; \left(\overrightarrow{a}^\overrightarrow{b}\right)=\left(\overrightarrow{a}^\overrightarrow{c}\right); \left(\overrightarrow{a}^\overrightarrow{d}\right)=\frac{\pi }{2}$ and $\left(\overrightarrow{a}^\overrightarrow{j}\right)$ is obtuse, then the value of $\left(x+y+z\right)$ is

Let $\overrightarrow{A}=2\overrightarrow{i}+\overrightarrow{k}, \overrightarrow{B}=\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$ and $\overrightarrow{C}=4\overrightarrow{i}-3\overrightarrow{j}+7\overrightarrow{k}$, then the vector $\overrightarrow{R}$, satisfying $\overrightarrow{R}\times \overrightarrow{B}=\overrightarrow{C}\times \overrightarrow{B}$ and $\overrightarrow{R}·\overrightarrow{A}=0,$ is $x\hat{i}+y\hat{j}+z\hat{k}$, then $\left(x+y+z\right)$ is equal to

Let $\overrightarrow{u}$ be a vector on rectangular coordinate system with sloping angle $60°$. Suppose that, $\left|\overrightarrow{u}-\hat{i}\right|$ is geometric mean of $\left|\overrightarrow{u}\right|$ and $\left|\overrightarrow{u}-2\hat{i}\right|$ where $\hat{i}$ is the unit vector along $x$-axis then $\left|\overrightarrow{u}\right|$ has the value equal to $\sqrt{a}-\sqrt{b}$ where $a, b\in N.$ Then the value of ${\left(a+b\right)}^3+{\left(a-b\right)}^3$ is