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.