Question

find the equation of a line passing through the points A(0,6,-9) and B(-3,-6,3). If D is the foot of perpendicular drawn from the point (7,4,-1) on AB, then find D and the equation of line CD

Answer 1

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Answer 2

Given:

Points A(0,6,-9) and B(-3,-6,3).

D is the foot of perpendicular drawn from the point C(7,4,-1) on AB

To Find:

D and the equation of line CD.

Solution:

Line AB can be represented as :

• 6j -9k + t ( 3i +  12j -12k ) = 6j - 9k + t ( i + 4j - 4k)

• Point D can be represented as : ( 3t , 6 + 12t, -9 -12t)
• Vector parallel to CD :
• (3t - 7) i + (12t +2)j   -  ( 12t + 8)k

CD is perpendicular to AB.

• CD.AB = 0 ( dot product )

• ((3t - 7) i + (12t +2)j   -  ( 12t + 8)k ) . (i +  4j -4k) = 0

• 3t - 7 + 48t + 8 + 48t + 32 = 0
• 99t = -33
• t = -1/3

Point D = (-1 , 2 , -5 ) .

Equation of CD = 7i + 4j -k + t ( 4i + j +2k)