MTH254-RecAct-Week4.pdf

# A 4 x 2 y 3 z 2 and 12 x 6 y 9 z 9 b x 3 y 2 z 4 and

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(a) 4 x - 2 y + 3 z = 2 and 12 x - 6 y + 9 z = 9. (b) x - 3 y + 2 z = - 4 and x - 3 z = 0. (c) 2 x + 3 y - 2 z = 7 and x - 2 y - 2 z = 2.

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Math 254 Week 4 Recitation Activity - Fall 2018 Page 2 of 2 (5) The purpose of this problem is to exercise our ability to visualize in three dimensions by exploring the relation between two lines. Consider two (infinite straight) lines, one containing the points A 1 = (1 , 2 , 3) and B 1 = (4 , 5 , 6) and the other containing the points A 2 = (7 , 8 , 10) and B 2 = (1 , 1 , 1) . (a) Find equations of the two lines described above. (Are these lines coplanar? i.e. Can the two lines “live” in the one plane?) (b) Find equations for two parallel planes so that each plane contains one of our lines. (i) Does the normal vector for either plane need to be orthogonal to the direc- tion vector for both lines? (Remember if a plane contains a line then the plane contains the direction vector of that line.) (ii) If needed, how can we create a vector orthogonal to both lines’ direction vectors? (iii) Could this vector created in (ii) be a normal vector for the parallel planes we want to build? (c) If we consider the shortest possible line segment that joins one line to the other, then this segment will be perpendicular to both lines. Why? (d) Is the distance between the two lines the same as the distance between the two parallel planes? (e) Calculate the distance between the lines. (Hint: use orthogonal projections.)
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• Fall '09
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