Question

If the direction ratios of a line are proportional to $1, - 3,2,$ then its direction cosines are:
A. $\dfrac{1}{{\sqrt {14} }},\dfrac{{ - 3}}{{\sqrt {14} }},\dfrac{2}{{\sqrt {14} }}$
B. $\dfrac{1}{{\sqrt {14} }},\dfrac{2}{{\sqrt {14} }},\dfrac{3}{{\sqrt {14} }}$
C. $\dfrac{{ - 1}}{{\sqrt {14} }},\dfrac{3}{{\sqrt {14} }},\dfrac{{ - 2}}{{\sqrt {14} }}$
D. $\dfrac{{ - 1}}{{\sqrt {14} }},\dfrac{{ - 2}}{{\sqrt {14} }},\dfrac{{ - 3}}{{\sqrt {14} }}$

Answer 1

Hint: In this question as direction ratio is given so to determine direction cosines, we use the correlation between direction ratios and direction cosines. Therefore, we use the formula of direction cosine where we can place the given value of the ratio.

Formula Used:
Directional cosines of a line segment which has direction ratio $a, b, c$ is:
\[ \pm \left( {\dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\dfrac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\dfrac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }}} \right)\]

Complete Step-by-step solution
It is given that the direction ratio of the line segment is proportional to $1, - 3,2$.
So, the direction cosine of the line will be given by using above mentioned formula, where $a = 1,b = - 3,c = 2$
Now, on putting the values of $a,b,c$ in the formula mentioned above, we get:
Directional cosines \[ = \pm \left( {\dfrac{1}{{\sqrt {{1^2} + {{( - 3)}^2} + {2^2}} }},\dfrac{{ - 3}}{{\sqrt {{1^2} + {{( - 3)}^2} + {2^2}} }},\dfrac{2}{{\sqrt {{1^2} + {{( - 3)}^2} + {2^2}} }}} \right)\]
\[ = \pm \left( {\dfrac{1}{{\sqrt {1 + 9 + 4} }},\dfrac{{ - 3}}{{\sqrt {1 + 9 + 4} }},\dfrac{2}{{\sqrt {1 + 9 + 4} }}} \right)\]
\[ = \pm \left( {\dfrac{1}{{\sqrt {14} }},\dfrac{{ - 3}}{{\sqrt {14} }},\dfrac{2}{{\sqrt {14} }}} \right)\]
So, we can say that the direction cosine can be \[ + \left( {\dfrac{1}{{\sqrt {14} }},\dfrac{{ - 3}}{{\sqrt {14} }},\dfrac{2}{{\sqrt {14} }}} \right)\]or\[\left( {\dfrac{{ - 1}}{{\sqrt {14} }},\dfrac{3}{{\sqrt {14} }},\dfrac{{ - 2}}{{\sqrt {14} }}} \right)\]

It can be written as \[ + \left( {\dfrac{1}{{\sqrt {14} }},\dfrac{{ - 3}}{{\sqrt {14} }},\dfrac{2}{{\sqrt {14} }}} \right)\] and \[\left( {\dfrac{{ - 1}}{{\sqrt {14} }},\dfrac{3}{{\sqrt {14} }},\dfrac{{ - 2}}{{\sqrt {14} }}} \right)\]

Hence the correct option can be either A and C.

Note: It is not necessary that a question can only have one answer. Like the given question has multiple answers because sometimes a value can be both positive and negative and to decide between them without any other information given is not possible. This question comes under the concept of directional cosines and vectors.