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This is my real name. Sorry for the inconvenience. We are three brothers using the same email so we all end up using different names for ourselves. We will just stick with one name now nonetheless.
I don't understand how to express or start off proving algebraically part a) and part b) ii). I posted this again because you said you had to get back to me about it.
And what is it exactly that they want you to sketch. Thanks for the help.

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 >

Questioner:Nick
Private:Yes  <<<<<<<<<<<<<<<<<<<<<<<< changed; no private questions.
Subject:Linear algebra 1
Question:

This is my real name. Sorry for the inconvenience. We are three brothers using the same email so we all end up using different names for ourselves.

>>>>>>>> Sure.

We will just stick with one name now nonetheless.

-------------------------------------------------------

(I hope the revision appears soon.)

I don't understand how to express or start off proving algebraically part a) and part b) ii). I posted this again because you said you had to get back to me about it.
And what is it exactly that they want you to sketch. Thanks for the help.

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 >

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#### Paul Klarreich

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