Question

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If A(4,8, 12), B(2, 4, 6), C(3,5, 4) and D(5,8,5) are four points, show that the lines
AB and CD intersect.​

Answer 1

Proved, lines AB and CD intersect

pt. of intersection is (1,2,3)

•Equation of line passing through

two points is:

(x - X1)/(X2-X1) = (y-Y1)/(Y2-Y1) =

(z-Z1)/(Z2-Z1) __________(1)

•Equation of line passing through

A(4,8, 12), B(2, 4, 6)

(x - 4)/(2-4) = (y-8)/(4-8) =(z-12)/(6-12)

(x - 4)/(-2) = (y-8)/(-4) =(z-12)/(-6)

•Equation of line passing through

C(3,5, 4), D(5, 8, 5)

(x - 3)/(5-3) = (y-5)/(8-5) =(z-4)/(5-4)

(x - 3)/2 = (y-5)/3 =(z-4)/1

•let a general points on both the

line

•For AB

(x - 4)/(-2) = (y-8)/(-4) =(z-12)/(-6) = k

x = 4-2k

y = 8-4k

z = 12-6k

•FOR CD

(x - 3)/2 = (y-5)/3 =(z-4)/1 = u

x = 3+2u

y = 5+3u

z = 4+u

•At point of intersection

4-2k = 3+2u

2k + 2u =1 ________(2)

8-4k = 5+3u

4k+3u = 3 ________(3)

12-6k = 4+u

6k +u = 8 ________(4)

•Solving 3 and 4 by elimination method

4k+3u = 3

18k +3u = 24

___________

14k = 21

k = 3/2

u = -1

•Now putting k &u in equation 2 if value of k&u satisfies equation 2 then these lines intersect otherwise lines don't intersect

2k + 2u =1

2(3/2) + 2(-1) = 1

3-2 = 1

1=1

•These lines intersect .

•point of intersection

x = 3+2u = 3+2(-1) = 3-2 = 1

y = 5+3u = 5 +3(-1) = 5-3 =2

z = 4+u = 4+(-1) = 4-1 = 3

•pt. of intersection is (1,2,3)