Question

If \[\lambda \left( 3\overset{\hat{\ }}{\mathop{i}}\,+2\overset{\hat{\ }}{\mathop{j}}\,-6\overset{\hat{\ }}{\mathop{k}}\, \right)\] is a unit vector, then the values of \[\lambda \]
(A) \[\pm \dfrac{1}{7}\]
(B) \[\pm 7\]
(C) \[\pm \sqrt{43}\]
(D) \[\pm \dfrac{1}{\sqrt{43}}\]

Answer 1

Hint: We are given a question based on vectors. We are given a unit vector with a character \[\lambda \] whose value we have to find using the given information. We will multiply the \[\lambda \] with the vector given and then we will calculate the magnitude of the resultant vector and equate it to 1. Then, solving further, we will get the value of \[\lambda \] for which the given vector is a unit vector.

Complete step by step answer:
According to the given question, we are given a unit vector and we are asked to find the value of \[\lambda \] in the given vector.
Unit vector is a vector of unit magnitude and direction.
\[\overset{\hat{\ }}{\mathop{a}}\,=\dfrac{a}{\left| a \right|}=1\] where ‘a’ is a unit vector
The vector that we have is,
\[\lambda \left( 3\overset{\hat{\ }}{\mathop{i}}\,+2\overset{\hat{\ }}{\mathop{j}}\,-6\overset{\hat{\ }}{\mathop{k}}\, \right)\]
We will now multiply \[\lambda \] to each of the terms within the brackets and we get,
\[\Rightarrow 3\lambda \overset{\hat{\ }}{\mathop{i}}\,+2\lambda \overset{\hat{\ }}{\mathop{j}}\,-6\lambda \overset{\hat{\ }}{\mathop{k}}\,\]
We will now calculate the magnitude of the above vector and equate it to 1, since we are given that it is a unit vector.
\[\left| 3\lambda \overset{\hat{\ }}{\mathop{i}}\,+2\lambda \overset{\hat{\ }}{\mathop{j}}\,-6\lambda \overset{\hat{\ }}{\mathop{k}}\, \right|=1\]
Calculating the magnitude now, we have,
\[\begin{align}
  & \Rightarrow \sqrt{{{\left( 3\lambda \right)}^{2}}+{{\left( 2\lambda \right)}^{2}}+{{\left( -6\lambda \right)}^{2}}}=1 \\
 & \Rightarrow \sqrt{9{{\lambda }^{2}}+4{{\lambda }^{2}}+36{{\lambda }^{2}}}=1 \\
 & \Rightarrow \sqrt{49{{\lambda }^{2}}}=1 \\
\end{align}\]
\[\begin{align}
  & \Rightarrow 7\lambda =1 \\
 & \Rightarrow \left| \lambda \right|=\dfrac{1}{7} \\
 & \Rightarrow \lambda =\pm \dfrac{1}{7} \\
\end{align}\]
Therefore, the value of \[\lambda =\pm \dfrac{1}{7}\]

So, the correct answer is “Option A”.

Note: We can also find the value of \[\lambda \] by finding the magnitude of the vector \[3\overset{\hat{\ }}{\mathop{i}}\,+2\overset{\hat{\ }}{\mathop{j}}\,-6\overset{\hat{\ }}{\mathop{k}}\,\] separately and then incorporate it with \[\lambda \] and hence find its value. While calculating the magnitude of the vector make sure that the position of the terms and the respective signs are carefully written else the answer will come out wrong.