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Question
I have no idea at all what they asking to be done in this problem. Thanks for any help, it is very appreciated.

Take a point P in R3 and two non-zero vectors v and w. Consider the vector equation
Q = P + s(vector v) + t(vector w) for any real s and t:
Let S be the set of all such points Q.
(a) Assume that vector v and vector w are parallel. Show algebraically that S is a line. Sketch of vector v, vector w,
P and the line S.
(b) Assume that vector v and vector w are not parallel.

i. Show that the following points of S are not colinear.
P, P +vector v, P + vector w

ii. For any of the points Q in S, show that
vector PQ is perpendicular to vector n = (vector v cross product vector w).
Remark. 1 Let pi be the plane through P and perpendicular to vector n. Part ii shows that S lies in pi. Part i showed that S is not a line. In fact, S is the plane pi.

iii. Sketch P, vector v, vector w, vector n and the plane S.

(c) Determine whether the following vector equations give you a point, a line or a plane.
i. (x, y, z) = (1, 4, 2) + s< 4,-5, 1 > +t< -8, 10, 3 >
ii. (x, y, z) = (-1, 0, 1) + s< -5, 10, 5 > + t< 3,-6,-3 >
iii. (x, y, z) = (5, 1, 2) + s< 0, 0, 0 > + t< 0, 0, 0 >

Answer
Billy, Timmy, Sam,

Wish you had more specific questions or could at least list some of the definitions and results presented to you in your course work. The subject of lines and planes in 3-D is covered pretty succinctly in various references; one example that you should look at is

‎science.kennesaw.edu/~plaval/math2203/linesplanes.pdf.

For the current problem, here are some results that should help you do your homework assignment:

If a vector V is parallel to vector W, then there components are proportional and one can write V = rW, where r is a scalar. Using this you can see that sV + tW = s(rW) + tW = (sr+t)W, which you can use for part a).

For part b), i) is easy to see since V and W are not colinear.

ii) 2 lines (non-colinear) define a plane, by definition the cross product of 2 vectors, say U = VxW, is a vector perpindicular to both of them, since V and W form a plane, any vector in the plane, i.e., PQ, must be perpindicular to U.

part c) The key to these questions is to determine if the 2 vectors given in the < > notation are parallel by determining if they are proportional to each other. For example, for ii) <-5,10,5> = (-3/5)<3,-6,-3>, so they are parallel.For iii) the 2 vectors are 0 so that the equation just gives a point. For i) it looks like the vectors are not parallel and so form a plane.

Good luck. BTW, do you know Julianne?

Randy

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