Question

If \[p=\overset{\hat{\ }}{\mathop{i}}\,+\overset{\hat{\ }}{\mathop{j}}\,,q=4\overset{\hat{\ }}{\mathop{k}}\,-\overset{\hat{\ }}{\mathop{j}}\,\ and\ r=\overset{\hat{\ }}{\mathop{i}}\,+\overset{\hat{\ }}{\mathop{k}}\,\], then the unit vector in the direction of $3p+q-2r$ is,
A. $\dfrac{1}{3}\left( \overset{\hat{\ }}{\mathop{i}}\,+2\overset{\hat{\ }}{\mathop{j}}\,+2\overset{\hat{\ }}{\mathop{k}}\, \right)$
B. $\dfrac{1}{3}\left( \overset{\hat{\ }}{\mathop{i}}\,-2\overset{\hat{\ }}{\mathop{j}}\,-2\overset{\hat{\ }}{\mathop{k}}\, \right)$
C. $\dfrac{1}{3}\left( \overset{\hat{\ }}{\mathop{i}}\,-2\overset{\hat{\ }}{\mathop{j}}\,+2\overset{\hat{\ }}{\mathop{k}}\, \right)$
D. $\overset{\hat{\ }}{\mathop{i}}\,-2\overset{\hat{\ }}{\mathop{j}}\,+2\overset{\hat{\ }}{\mathop{k}}\,$

Answer 1

Hint: We will be using the concept of vector to solve the problem. We will using the concept that a unit vector in direction of a vector $\overrightarrow{a}\ is\ \dfrac{\overrightarrow{a}}{\left| \overrightarrow{a} \right|}$ where $\left| \overrightarrow{a} \right|$ is the magnitude of the vector.

Complete step-by-step answer:
Now, we have been given that,
\[\begin{align}
  & p=\overset{\hat{\ }}{\mathop{i}}\,+\overset{\hat{\ }}{\mathop{j}}\,...........\left( 1 \right) \\
 & q=4\overset{\hat{\ }}{\mathop{k}}\,-\overset{\hat{\ }}{\mathop{j}}\,\ .........\left( 2 \right) \\
 & r=\overset{\hat{\ }}{\mathop{i}}\,+\overset{\hat{\ }}{\mathop{k}}\,............\left( 3 \right) \\
\end{align}\]
We have to find a unit vector in the direction of $3p+q-2r$. So, we have using (1), (2) and (3).
$\begin{align}
  & 3p+q-2r=3\left( \overset{\hat{\ }}{\mathop{i}}\,+\overset{\hat{\ }}{\mathop{j}}\, \right)+4\overset{\hat{\ }}{\mathop{k}}\,-\overset{\hat{\ }}{\mathop{j}}\,-2\left( \overset{\hat{\ }}{\mathop{i}}\,+\overset{\hat{\ }}{\mathop{k}}\, \right) \\
 & =3\overset{\hat{\ }}{\mathop{i}}\,+3\overset{\hat{\ }}{\mathop{j}}\,+4\overset{\hat{\ }}{\mathop{k}}\,-\overset{\hat{\ }}{\mathop{j}}\,-2\overset{\hat{\ }}{\mathop{i}}\,-2\overset{\hat{\ }}{\mathop{k}}\, \\
 & =\overset{\hat{\ }}{\mathop{i}}\,+2\overset{\hat{\ }}{\mathop{j}}\,+2\overset{\hat{\ }}{\mathop{k}}\, \\
 & =\left( \overset{\hat{\ }}{\mathop{i}}\,+2\overset{\hat{\ }}{\mathop{j}}\,+2\overset{\hat{\ }}{\mathop{k}}\, \right) \\
\end{align}$
Now, we have to find $\left| 3p+q-2r \right|$. We know,
$\left| a\overset{\hat{\ }}{\mathop{i}}\,+b\overset{\hat{\ }}{\mathop{j}}\,+c\overset{\hat{\ }}{\mathop{k}}\, \right|=\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}$
So, we have,
$\begin{align}
  & \left| \overset{\hat{\ }}{\mathop{i}}\,+2\overset{\hat{\ }}{\mathop{j}}\,+2\overset{\hat{\ }}{\mathop{k}}\, \right|=\sqrt{1+4+4} \\
 & =\sqrt{9} \\
 & =3 \\
\end{align}$
Now, we know that unit vector along the direction of a vector $\overrightarrow{a}\ is\ \dfrac{\overrightarrow{a}}{\left| \overrightarrow{a} \right|}$.
So, we have unit vector along $3p+q-2r$ as,
$\begin{align}
  & \dfrac{\left( \overset{\hat{\ }}{\mathop{i}}\,+2\overset{\hat{\ }}{\mathop{j}}\,+2\overset{\hat{\ }}{\mathop{k}}\, \right)}{3} \\
 & =\dfrac{1}{3}\left( \overset{\hat{\ }}{\mathop{i}}\,+2\overset{\hat{\ }}{\mathop{j}}\,+2\overset{\hat{\ }}{\mathop{k}}\, \right) \\
\end{align}$
Hence, the correct option is (A).

Note: To solve these type of questions it is important to note that a unit vector along a vector $\overrightarrow{a}\ is\ \dfrac{\overrightarrow{a}}{\left| \overrightarrow{a} \right|}$. Also to find the magnitude of vector $\left| x\overset{\hat{\ }}{\mathop{i}}\,+y\overset{\hat{\ }}{\mathop{j}}\,+z\overset{\hat{\ }}{\mathop{k}}\, \right|=\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}$.