Question

What are the direction cosines of the joins of the following pairs of points (3, -4, 7), (0, 2, 5)
A). \[\dfrac{3}{7},\dfrac{{ - 6}}{7},\dfrac{2}{7}\]
B). \[\dfrac{3}{7},\dfrac{6}{7},\dfrac{{ - 2}}{7}\]
C). \[\dfrac{{ - 3}}{7},\dfrac{{ - 6}}{7},\dfrac{{ - 2}}{7}\]
D). \[\dfrac{3}{7},\dfrac{6}{7},\dfrac{2}{7}\]

Answer 1

Hint: First we will find the direction ratios. Direction ratio is the difference between the two coordinates of each coefficient. Then, we will find the direction cosines using its formula.

Complete step-by-step solution:
We will let the points from first pair as ${x_1} = 3,{y_1} = - 4,{z_1} = 7$ and from second pair as ${x_2} = 0,{y_2} = 2,{z_2} = 5$.
Now, we will find the direction ratios of the points. Direction ratios of the points are the difference of the two coordinates of each coefficient.
We will assume the first direction ratio as ‘a’.
$a = {x_1} - {x_2}$
Substituting the values in the equation.
$a = 3 - 0$
$a = 3$
We will assume the second direction ratio as ‘b’.
$b = {y_1} - {y_2}$
Substituting the values in the equation.
$b = ( - 4) - 2$
$b = - 6$
We will assume the third direction ratio as ‘c’.
$c = {z_1} - {z_2}$
Substituting the values in the equation:
$c = 7 - 5$
$c = 2$
Now, we will find the value of direction cosines. The formula of finding direction cosines is: $\dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\dfrac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\dfrac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$. Here, a, b and c are the direction ratios.
Substituting the values in the formula
$\dfrac{3}{{\sqrt {{3^2} + {{( - 6)}^2} + {2^2}} }},\dfrac{{ - 6}}{{\sqrt {{3^2} + {{( - 6)}^2} + {2^2}} }},\dfrac{2}{{\sqrt {{3^2} + {{( - 6)}^2} + {2^2}} }}$
$\dfrac{3}{{\sqrt {9 + 36 + 4} }},\dfrac{{ - 6}}{{\sqrt {9 + 36 + 4} }},\dfrac{2}{{\sqrt {9 + 36 + 4} }}$
$\dfrac{3}{{\sqrt {49} }},\dfrac{{ - 6}}{{\sqrt {49} }},\dfrac{2}{{\sqrt {49} }}$
Value of $\sqrt {49} $is 7. So, substituting it in the above equation.
$\dfrac{3}{7},\dfrac{{ - 6}}{7},\dfrac{2}{7}$
The value of direction cosines is $\dfrac{3}{7},\dfrac{{ - 6}}{7},\dfrac{2}{7}$.
So, option (A) is the correct answer.

Note: In these types of questions where 2 points are given, we have to find the direction ratios separately. If only one point will be given in the question, then, we will assume those points to be the direction ratios. And solve the questions accordingly using the same formulas as above.