Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 1: KCET (UG) 2019

If the vectors $a\hat{i}+\hat{j}+\hat{k}, \hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ are coplanar $\left(a\ne b\ne c\ne 1\right)$, then the value of $abc-\left(a+b+c\right)=$

If $\overrightarrow{a}=\hat{i}+\lambda \hat{j}+2\hat{k} \& \overrightarrow{b}=\mu \hat{i}+\hat{j}-\hat{k}$ are orthogonal and $\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|$, then $\left(\lambda ,\mu \right)=$

The diagonals of a parallelogram are the vectors $3\hat{i}+6\hat{j}-2\hat{k}$ and $-\hat{i}-2\hat{j}-8\hat{k}$ then, the length of the shorter side of parallelogram is

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are mutually perpendicular unit vectors, then $\left(3\overrightarrow{a}+2\overrightarrow{b}\right)·\left(5\overrightarrow{a}-6\overrightarrow{b}\right)=$

A vector $\overrightarrow{a}$ makes equal acute angles on the coordinate axis. Then the projection of vector $\overrightarrow{b}=5\hat{i}+7\hat{j}+\hat{k}$ on $\overrightarrow{a}$ is

If $\overrightarrow{a}·\overrightarrow{b}=0$ and $\overrightarrow{a}+\overrightarrow{b}$ makes an angle $60°$ with $\overrightarrow{a}$ then 

The area of the quadrilateral $ABCD,$ where $A(0, 4, 1), B(2, 3,-1), C(4, 5, 0)$ and $D(2, 6, 2)$ is equal to

If the area of the parallelogram with $\overrightarrow{a}$ and $\overrightarrow{b}$ as two adjacent sides is $15$ sq.  units then the area of the parallelogram having $3\overrightarrow{a}+2\overrightarrow{b}$ and $\overrightarrow{a}+3\overrightarrow{b}$ as two adjacent sides in sq. units is