Vectors and Dot Products: Understanding and Applying Vector Methods

[imsoconfused]

Vectors, dot products help!

OK, I know this is probably really annoying but I have a ton of questions. (I would go to the math tutor but they aren't here yet because the semester just started.)

1. Find the angle POQ by vector methods if P=<1,1,0>, O=<0,0,0>, and Q=<1,2,-2>.

What I have tried to use is cos(theta)=a.b/ab, but since there are three vectors I'm having trouble figuring out how to apply the theorem. The thing that confuses me most is O=<0,0,0>, it's not really a vector but somehow I'm supposed to use it to calculate an angle. The professor told us the answer is 45degrees, I got 20degrees.

2. Describe all points (x,y) such that v=xi+yj satisfies:
(a) |v|= 2
(b) |v-1|= 2
(c) v.i = 2
(d) v.i=|v|

This one really confuses me. I've tried just plugging the given v's into the equation, but the answers the professor gave us seem so far away from that I'm pretty certain that that's not the way to go about doing it. What'd be helpful for me would be the worked out solution to one and then I'll apply it to the other three.

3. Find the angle between the diagonal of cube and (a) an edge (b) the diagonal of the face and (c) another diagonal of the cube.

I think I somewhat understand how to get (a) and (b), but (c) makes absolutely no sense to me. I've drawn out a cube, and I've used unit vectors to find the angles of the first two, but I can't figure out what I'm supposed to be solving for for (c).

4. Suppose I=(i+j)/sqrt(2) and J=(i-j)/sqrt(2). Check I.J=0 and write A=2i+3j as a combination aI+bJ. (a=A.I and b=A.J, solved in another problem.)

I don't even know what this one is asking for. I have an answer key for this one, but it doesn't explain what it's doing and I can't see what it is. Here is the answer, I'd love it if someone would show me what it means.

A = 2 i + 3 j = sqrt(2)(I+J)+(3sqrt(2)/2)(I-J)=aI+bJ with a=sqrt(2)+(3sqrt(2))/2 and b=sqrt(2)-(3sqrt(2))/2

5. If |A+B|sq = |A|sq + |B|sq, prove that A is perpendicular to B.

I assume this means I have to find a way to use the dot product to show that A.B=0, but I don't know where to begin. Would I use cos(theta)=(A.B)/(|A||B|)?

6. (Last one!) (a) verify the Schwarz inequality |V.W|_< |V||W| for V=i+2j+2k and W= 2i+2j+k. (b) What does the inequality become when V=(sqrt(x), sqrt(y)) and W=(sqrt(y), sqrt(x))?

This one seems a little more manageable, but I'm still at a loss as to how to begin. Do I just go right in and plug in vectors? The answer is 8_< 3.3 and 2(sqrt(xy))_<x+y.

I know this is a lot, it's one night's homework and I'm not terribly clever at math. The text I'm using is Strang which has about zero examples, and I will appreciate any and all advice (or answers!).
Thanks so much!

 

[salman213]

well just to get u started on the first one. 0,0,0 is basically being used as a point not reall vector. When you make a triangle and you want to find the angle between two sides you say find the angle <ABC for example. So its similar in that case. so for the first one all you have to do is find the angle between the two vectors.

Dot product of P and Q = magnitudeofP*MagnitudeQ*Cos(ANGLE U WANT)

you should get 45..

i got angle = cos-1(1/sqrt2) = 45

 

[imsoconfused]

yay that seems to be correct! I suppose I made a calculation error the first time I tried it without using O, so I got confused when I tried to use it as a vector.

 

[Defennder]

Seems like questions 1-5 is from the pre-calc forum.

6) Part a) just requires you to plug in the given vectors v and w.
b)It's the same as a), just need to replace the numbers with sqrt(x) and sqrt(y).