Question

The direction cosines of a line segment AB are $- \dfrac{2}{{\sqrt {17} }},\dfrac{3}{{\sqrt {17} }}, - \dfrac{2}{{\sqrt {17} }}$ . If AB $= \sqrt {17}$ and the coordinates of A are $(3, - 6,10)$ , then the coordinates of B are
A.(1, -2, 4)
B.(2, ,5, 8)
C.(-1,3,-8)
D.(1, -3, 8)

Answer 1

Hint : Direction cosines of a line are the cosines of the angel made by the line with positive direction of the coordinate axes. Direction ratios of a line are the numbers which are proportional to the direction cosine line. Let’s take the coordinate of B be $(x,y,z)$ . Since coordinates of A are given and using the given data we can solve this.

Complete step-by-step answer :
Given,
The direction cosines of a line segment AB $= \left( { - \dfrac{2}{{\sqrt {17} }},\dfrac{3}{{\sqrt {17} }}, - \dfrac{2}{{\sqrt {17} }}} \right)$ .
Since AB $= \sqrt {17}$ .
By the definition of direction ratios we have, direction ratios of a line are the number which are proportional to the direction cosine line.
$\therefore$ Direction ratio of the line AB $= ( - 2,3 - 2)$
We know, direction ratio of the line AB= Coordinates of B Coordinates of A.
Let the coordinate of B is $(x,y,z)$ . Then we have,
$( - 2,3, - 2) = (x,y,z) - (3, - 6,10)$
$\Rightarrow ( - 2,3, - 2) = (x - 3,y + 6,z - 10)$
Equating the corresponding values we get,
$\Rightarrow x - 3 = - 2,$ $y + 6 = 3$ , $z - 10 = - 2$ .
Solving each we get,
$x = - 2 + 3$
$\Rightarrow x = 1$
$y = 3 - 6$
$\Rightarrow y = - 3$
$z = - 2 + 10$
$\Rightarrow z = 8$
That is the coordinate of B is $(x,y,z) = (1, - 3,8)$ ,
So, the correct answer is “Option D”.

Note : If we find the distance between A $(3, - 6,10)$ and B $(1, - 3,8)$ we will get AB $= \sqrt {17}$ . If we get a different value then our obtained coordinate of B is wrong. Using this we can check whether the obtained answer is correct or not. Follow the same procedure for any different values of direction cosines and coordinate of A.