Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 3: EXERCISE-3

Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $30°$. The area of face $ABC$ is $120$ sq. units and the area of face $BCD$ is $80$ sq. units and $BC=10$ units, then the volume of tetrahedron is

In $∆ABC$, Let $B\left(3,3,-9\right)$ and $AB=AC$ and altitude $AD$ is $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{3}$. If coordinates of $C$ are $\left(\alpha ,\beta ,\gamma \right)$, then $\left|\alpha \right|+\left|\beta \right|+\left|\gamma \right|$ is

$\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ 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 $\overrightarrow{u}$ 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

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

$ABCD$ is a tetrahedron with p.v.'s of its angular point as $A\left(-5, 22, 5\right); B\left(1, 2, 3\right); C\left(4, 3, 2\right)$ and $D\left(-1, 2, -3\right).$ If the area of the triangle $AEF$ where the quadrilaterals $ABDE$ and $ABCF$ are parallelograms is $\sqrt{S},$ then the value of $S$ is