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Planes & Vectors : Points and Equations of Parallel Lines

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4) Line L is defined by the following equation:
Line M is defined by:
Find a and b, if known that lines are parallel to each other.

5) Line L is defined by intersection of two planes
2x+3y?z = 1 and -2x+y+2z = 0
Plane P is defined by -2x + y ?z = 4
Find any directional vector for the line, point of intersection of line and plane

6) Points A(0,1,3), B(-l, 3, 7), C( -4, 1, 0) and D(a, -1, 0). For which value of a will all four points lie in the same plane?

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Points are found that allow lines to be parallel or in the same plane. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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4. A standard form for the equations of a line is:

x - x0 y - y0 z - z0
------ = ------ = ------
a b c

Here (x0, y0, z0) is a point on the line, and the numbers a, b, and c determine the direction along the line: the vector a*I = b*J + c*K is parallel to the line.
For the first line, i = 3j - 5k, or v1 = i - 3j + 5k
For the second line, 3i = bj + ak, or v2 = 3i - bj - ak
If these two vectors are parallel to each other, then v1 x v2 = ...

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