Consider the parallel lines L defined by x=6+6t, y=2t, z=-3t and M defined by x=

Question

Consider the parallel lines L defined by x=6+6t, y=2t, z=-3t and M defined by x=6t, y=-4+2t, and z=4-3t.

Find an equation of the plane which contains L and M. Hint: Choose a point P on L and distinct points Q and R on M, and then find an equation of the plane that passes through P, G, and R.

I don't need help with the whole problem, I just have one issue that I cannot solve. The answer given by the teacher for the points are P: (6,0,0), Q: (0,-4,4), and R: (6,-2,1). I understand how to get points P and Q, but I do not understand the method of finding point R. Is it the intersection of the parametric equations? An explanation along with a written solution to it wold be great.

Explanation / Answer

parallel lines never intersect .

your teacher puts t=0 in  x=6+6t, y=2t, z=-3t to get P(6,0,0)

t=0 in  x=6t, y=-4+2t, and z=4-3t. to get Q(0,-4,4)

t=1 in  x=6t, y=-4+2t, and z=4-3t. to get R(6,-2,1)

take general point X(x,y,z)

to find equation of plane

find PX,PQ,PR,PQxPR

then equate dot product of PQxPR and PX equal to zero

(PQxPR).PX =0

then you get equation of plane

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