Question

A vector $ \overrightarrow r $ is equally inclined with the coordinate axes. If the tip of $ \overrightarrow r $ is in positive octant and $ \left| {\overrightarrow r } \right| = 6 $ , then $ \overrightarrow r $ is
A. $ 2\sqrt 3 (\widehat i - \widehat j + \widehat k) $
B. $ 2\sqrt 3 ( - \widehat i + \widehat j + \widehat k) $
C. \[2\sqrt 3 (\widehat i + \widehat j - \widehat k)\]
D. $ 2\sqrt 3 (\widehat i + \widehat j + \widehat k) $

Answer 1

Hint: When any vector is equally inclined with coordinate axes, it makes equal angles with all three coordinate axes.So their direction cosines will also be equal.
Generally we write DC’s as l, m and n
We will use cosines of equally inclined vector as $ (\dfrac{1}{{\sqrt 3 }},\dfrac{1}{{\sqrt 3 }},\dfrac{1}{{\sqrt 3 }}) $
Then we will use a formula to find the vector $ \overrightarrow r $ . Which will give us the final answer for this question.
Formula to find vector $ \overrightarrow r $ ,
\[\overrightarrow r = \left| {\overrightarrow r } \right|(l\widehat i + m\widehat j + n\widehat k)\]

Complete step-by-step answer:
Given,
 $ \left| {\overrightarrow r } \right| = 6 $
We need to find , $ \overrightarrow r $
So first we will find direction cosines of vector $ \overrightarrow r $ ,
Let l, m and n are DC's of $ \overrightarrow r $
Then ,
 $ l = m = n = 1 $ ( as it is already explained in hint that $ \overrightarrow r $ will have equal DC's)
Now we can write,
 $ {l^2} + {m^2} + {n^2} = 1 $
And all are equal so we can say that,
 $ 3{l^2} = 1 $
Or,
 $ {l^2} = \dfrac{1}{3} $
Or
 $ l = \pm \dfrac{1}{{\sqrt 3 }} $
It is given in question that tip if vector is in positive octant so we will take positive value of l
So ,
Now we have values of l ,m and n as $ \dfrac{1}{{\sqrt 3 }} $
We have formula for vector $ \overrightarrow r $ as,
\[\Rightarrow \overrightarrow r = \left| {\overrightarrow r } \right|(l\widehat i + m\widehat j + n\widehat k)\]
And value of $ \overrightarrow r $ is given as, $ \left| {\overrightarrow r } \right| = 6 $
Now in this Step we will substitute values in formula,
So,
\[\Rightarrow \overrightarrow r = 6(\dfrac{1}{{\sqrt 3 }}\widehat i + \dfrac{1}{{\sqrt 3 }}\widehat j + \dfrac{1}{{\sqrt 3 }}\widehat k)\]
Or
By taking $ \dfrac{1}{{\sqrt 3 }} $ common
\[\overrightarrow r = 6 \times \dfrac{1}{{\sqrt 3 }}(\widehat i + \widehat j + \widehat k)\]
After simplifying we will get,
 $ \overrightarrow r =2\sqrt 3 (\widehat i + \widehat j + \widehat k) $

So, the correct answer is “Option D”.

Note: 1. While calculating direction cosines it is to be remembered that we are taking all the angles in an anticlockwise direction.
2. When DC's are not known then we can find them by this formula
 $ \operatorname{Cos} a = \dfrac{x}{{\left| {\overrightarrow r } \right|}} $ , $ \operatorname{Cos} b = \dfrac{y}{{\left| {\overrightarrow r } \right|}} $ and \[\operatorname{Cos} c = \dfrac{z}{{\left| {\overrightarrow r } \right|}}\]