find the equation of line passing through the point (1,-1,1) and perpendicular to the lines joining the points (4,3,2), (1,-1,0) - Maths - Three Dimensional Geometry

Question

find the equation of line passing through the point (1,-1,1) and
perpendicular to the lines joining the points (4,3,2), (1,-1,0) and
(1,2,-1), (2,1,1)

Ajanta Trivedi · Accepted Answer

let the given points be A(4,3,2), B(1,-1,0) and C(1,2,-1),D(2,1,1).

the direction ratios of the line AB= $(1 - 4, - 1 - 3, 0 - 2) \equiv (- 3, - 4, - 2)$

the direction ratios of the line CD= $(2 - 1, 1 - 2, 1 + 1) \equiv (1, - 1, 2)$

let the direction ratios of the desired line be (a,b,c).

and the desired line is perpendicular to AB and CD. therefore

$- 3 a - 4 b - 2 c = 0; \Rightarrow 3 a + 4 b + 2 c = 0.... (1) a - b + 2 c = 0... (2)$

let c=1 the equations become 3a+4b+2=0..1(b) and a-b+2=0...2(b)

by solving 1(b) and 2(b) we get $a = - \frac{10}{7} a n d b = \frac{4}{7}$

therefore direction ratios of desired line is $(- \frac{10}{7}, \frac{4}{7}, 1) \equiv (- 10, 4, 7)$

since the line passes through the point (1,-1,1), the equation of the desired line is :

$\frac{x - 1}{- 10} = \frac{y + 1}{4} = \frac{z - 1}{7}$

find the equation of line passing through the point (1,-1,1) and perpendicular to the lines joining the points (4,3,2), (1,-1,0) and (1,2,-1), (2,1,1)