M L Aggarwal Solutions for Chapter: Vectors, Exercise 3: EXERCISE

Write the value of: $\left(\hat{i}\times \hat{j}\right)·\hat{k}+\hat{i}·\hat{j}.$

Write the value of: $\left(\hat{j}\times \hat{k}\right)·\hat{i}+\left(\hat{i}\times \hat{k}\right)·\hat{j}+\left(\hat{i}\times \hat{j}\right)·\hat{k}.$

If $\overrightarrow{a}=4\hat{i}+5\hat{j}-\hat{k}$, $\overrightarrow{b}=\hat{i}-4\hat{j}+5\hat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{j}-\hat{k}$, find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{c}$ and $\overrightarrow{b}$ and $\overrightarrow{d}·\overrightarrow{a}=21.$

(Answer in the form $a\hat{i}+b\hat{j}+c\hat{k}$)

In $∆OAB, \overrightarrow{OA}=3\hat{i}+2\hat{j}-\hat{k}$and $\overrightarrow{OB}=\hat{i}+3\hat{j}+\hat{k}$. Find the value of $k$, if the area of the triangle $OAB$ is $\frac{3}{2}\sqrt{k}$

Find the value of $k$, if the area of the triangle formed by the points $A\left(1, 2, 3\right), B\left(2, -1, 4\right)$ and $C\left(4, 5, -1\right)$ is $\frac{\sqrt{k}}{2}$ sq. units

Find the value of $k$, if the area of a parallelogram whose diagonals are determined by the vectors $\overrightarrow{a}=3\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{b}=\hat{i}-3\hat{j}+4\hat{k}$ is $5\sqrt{k}$ sq. units

The vectors $-2\hat{i}+4\hat{j}+4\hat{k}$ and $-4\hat{i}-2\hat{k}$ represent the diagonals $BD$ and $AC$ of a parallelogram $ABCD$. Find the value of $k$, if the area of the parallelogram is $6\sqrt{k}$ sq. units

The two adjacent sides of a parallelogram are represented by the vectors $2\hat{i}-4\hat{j}-5\hat{k}$ and $2\hat{i}+2\hat{j}+3\hat{k}$. Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.