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Question
I have no idea what to do. Any help would be greatly 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 >

(a) If the vectors are parallel, then they are the same line when taken from a fixed point P.
Since this is so, it doesn't matter what multiples of which one are used, for either one puts the points on the same line.

(b) If the vectors v and w are not parallel, then the point could be anywhere in the plane extending from point P in directions v and w.
d
i. If we take P as the origin, v=(1,0,0), and w = (0,1,0), to be co-linear, all combinations of these two vectors should be on a line.  Take the three point P, P+v, and P+w.  Clearly, if two points are taken to define a line, the third point is not on the line.

ii. It is known that the cross product of two vectors gives the vector that is perpendicular to both.  That means any constants could be chosen and a perpendicular vector would be found.  This implies that S lies in a plane.

(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 >
defines a plane since the two vectors are not quite multiples of each other.
If the vector times s is multiplied by -2, the vector <-8,10,-2> is gotten,
and this is different from <-8,-10,3>.

ii. (x, y, z) = (-1, 0, 1) + s< -5, 10, 5 > + t< 3,-6,-3 > forms a line, since <-5,10,5>
and <3,-6,-3> are both multiples of <1,-2,-1> ( 1st is by -5, 2nd is by 3).

iii. (x, y, z) = (5, 1, 2) + s< 0, 0, 0 > + t< 0, 0, 0 > defines a point since no matter what the values of s and t are, there is nothing added on.

Volunteer

Scott A Wilson

Expertise

I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

Experience

Experience in the area; I have tutored people in the above areas of mathematics for over two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

Publications
My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

Education/Credentials
Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

Past/Present Clients
My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.