Question

If $\overrightarrow{a}$ is a non-zero vector of magnitude $a$ and $\lambda \overrightarrow{a}$ is a unit vector, then find the value of $\lambda .$

Answer 1

Hint : If we have a unit vector $\widehat{b}$ then it’s value is given by the following expression $\widehat{b}=\dfrac{\overrightarrow{b}}{\left| \overrightarrow{b} \right|}$ .
where $\left| \overrightarrow{b} \right|$ is the magnitude of $\overrightarrow{b}$ ,
Using the above equation and information given in the question we can find out the value of $\lambda$ .

Complete step-by-step answer :
As we know value of a unit vector $\widehat{a}$ is given by,
$\widehat{a}=\dfrac{\overrightarrow{a}}{\left| \overrightarrow{a} \right|}......(1)$
Where $\left| \overrightarrow{a} \right|$ is the magnitude of $\overrightarrow{a}$ .
For example, suppose
$\overrightarrow{a}=2\widehat{i}+2\overset\frown{j}+\widehat{k}$
where $\overset\frown{i},\overset\frown{j},\overset\frown{k}$ are the unit vectors in the direction $x,y\,and\,z$ respectively
so, to find the magnitude of $\overrightarrow{a}$ i.e. $\left| \overrightarrow{a} \right|$ we need to add the individual squares of the coefficients of the unit vectors and take the square root of the whole expression so, $\left| \overrightarrow{a} \right|$ would be equal to $\sqrt{{{2}^{2}}+{{2}^{2}}+{{1}^{2}}}$
hence, $\left| \overrightarrow{a} \right|=\sqrt{{{2}^{2}}+{{2}^{2}}+{{1}^{2}}}=\sqrt{4+4+1}=\sqrt{9}=3$ and hence,
$\widehat{a}=\dfrac{\overrightarrow{a}}{\left| \overrightarrow{a} \right|}=\dfrac{2\widehat{i}+2\overset\frown{j}+\widehat{k}}{3}$
Now, moving back to the given question,
We are given that:
$\widehat{a}=\lambda \overrightarrow{a}......(2)$
Comparing and equating both the equations (1) and (2) we get,
$\lambda \widehat{a}=\dfrac{\overrightarrow{a}}{\left| \overrightarrow{a} \right|}$
Now, by dividing both sides by $\overrightarrow{a}$ , we get
$\lambda =\dfrac{1}{\left| \overrightarrow{a} \right|}$
Hence value of $\lambda$ is equal to $\lambda =\dfrac{1}{\left| \overrightarrow{a} \right|}$ .

Note : To solve this problem you need to know the basics of vectors and unit vectors.
You should have a good grasp over what is a unit vector and what is it’s magnitude which is understood by its name (i.e. unit) that is 1. Always remember the following equation of unit vectors for future references:
If we are given a vector $\overrightarrow{b}$ , then suppose $\overset\frown{b}$ is the unit vector $\overrightarrow{b}$ and it’s value will be equal to $\dfrac{\overrightarrow{b}}{\left| \overrightarrow{b} \right|}$ , i.e. $\widehat{b}=\dfrac{\overrightarrow{b}}{\left| \overrightarrow{b} \right|}$ where $\left| \overrightarrow{b} \right|$ is the magnitude of $\overrightarrow{b}$ .