Question

The direction cosines of the line joining the points $(2,3, - 1)$ and $(3, - 2,1)$are
A.$- 1,5, - 2$
B.$\dfrac{1}{{\sqrt {30} }}, - \sqrt {\dfrac{5}{6}} ,\sqrt {\dfrac{2}{{15}}}$
C.$- \dfrac{1}{{30}},\dfrac{1}{6}, - \dfrac{1}{{15}}$
D.None of these above

Answer 1

Hint: To find the direction cosines, we first need to find the direction ratios. Direction ratios are the difference between the two coordinates of each coefficient.

Complete step-by-step answer:
Given in the question, the points $(2,3, - 1)$and $(3, - 2,1)$.
We let the points be

$${x_1} = 2,{y_1} = 3,{z_1} = - 1 \\\ and \\\ {x_2} = 3,{y_2} = - 2,{z_2} = 1 \\\$$

Now, we will find the direction ratios of the points, which are
${x_2} - {x_1} = 3 - 2 = 1$ with respect to x
${y_2} - {y_1} = - 2 - 3 = - 5$ with respect to y
${z_2} - {z_1} = 1 - ( - 1) = 2$ with respect to z
To find the direction cosines of line joining the points $(2,3, - 1)$ and $(3, - 2,1)$, we will have to use the formulas $\dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\dfrac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\dfrac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$where a, b, c are the respective direction ratios.

$$\dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }} \\\ = \dfrac{1}{{\sqrt {{1^2} + {{\left( { - 5} \right)}^2} + {2^2}} }} \\\ = \dfrac{1}{{\sqrt {1 + 25 + 4} }} \\\ = \dfrac{1}{{\sqrt {30} }} \\\$$ $$\dfrac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }} \\\ ,=\dfrac{{ - 5}}{{\sqrt {{1^2} + {{\left( { - 5} \right)}^2} + {2^2}} }} \\\ ,=\dfrac{{ - 5}}{{\sqrt {1 + 25 + 4} }} \\\ ,=\dfrac{{ - 5}}{{\sqrt {30} }} = - \sqrt {\dfrac{5}{6}} \\\$$ $$\dfrac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }} \\\ , = \dfrac{2}{{\sqrt {{1^2} + {{\left( { - 5} \right)}^2} + {2^2}} }} \\\ , = \dfrac{2}{{\sqrt {1 + 25 + 4} }} \\\ ,= \dfrac{2}{{\sqrt {30} }} = \sqrt {\dfrac{2}{{15}}} \\\$$

Therefore, the direction cosines of the line joining the points $(2,3, - 1)$ and $(3, - 2,1)$are $\dfrac{1}{{\sqrt {30} }}, - \sqrt {\dfrac{5}{6}} ,\sqrt {\dfrac{2}{{15}}}$.
Thus, the answer is option B.

Note: In these types of questions where two points are given, we have to calculate the direction ratios separately. If only one point would be given, then we will assume the coordinates of the point to the direction ratio itself and proceed to further calculation in the same way as above.