Question

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

Answer:

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Step-by-step explanation:

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.

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)