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This is really long, but the questions I have really are rudimentary at best.

I have seven total questions.

You're given two vectors that only have an x and y component, A, and B, and the positive Z axis is out of the page.

A points roughly (no numbers given) 45 degrees (north-east). B point roughly 315 degrees (north-west).

I've tried to read the class notes on vectors and things, however I really can't seem to understand it.

There are basically three things I'm trying to understand.

Vectors, which I understand, and the difference between cross products and dot products.

Cross products seem to be |a||b|sine([itex]\theta[/itex]).

Dot products seem to be abCosine([itex]\theta[/itex]).

So please explain if what I'm doing is right-

24

What direction is vector

**axb**? (A times B)

So since it's a cross product, I guess it's the absolute magnitude of a and b times sine.

However I really don't understand the angle thing. How is the angle measured? Is the [itex]\theta[/itex] measured depending on the x or y axis? Or just THE angle between A and B?

So I wrote down the direction is upwards with no x-component? I have no better guess. Approximately the angle between a and b looks to be 90-120 degrees, 90 degrees, which would appear to sort of "add" a and b, just with larger magnitude.

25

The next question is how is bxa related to axb in magnitude and direction?

I'm sure that I'm wrong, however i can only think that cross products are commutative and that they are both identical. I guess there could be a thing with the angle, but still I'm not sure.

26

How could you change the direction of a, leaving it in the x, y plane to make axb zero?

Sine(180)=0, so you would just make a paralell to B.

27

If we rotate the coordinate axis so that x points towards the bottom of the paper, which I assume means to rotate the grid by 90 degrees clockwise, "how will axb change?"

This really doesn't seem to be relevant? I'm supposing a and b don't move with the graph, but either ways since first of all cross products are absolute values and that the angle between a and b is the same no matter how you rotate the xy axis, 27 seems to be a pointless question.

I think I sort of understand dot products after reading the class notes.

So one question I have is the result of a dot product, such as a[itex]\bullet[/itex]b, is C, which is a scalar. So a scalar is just a number, but I'm confused what the difference between a scalar and a vector is. A scalar has an angle and a magnitude, I think, so isn't that just a vector?

And the only way to get a negative dot product is with the angle>90, right? Since a and b are both magnitudes, which cannot be negative?

28. How could you change the direction of a, leaving it in the xy plane to make a dot b as large as possible?

Since cosine 0 or 180= maximum, making A to be parallel would give the largest dot product.

29

How could you change a's direction to make a dot b as small as possible? Can c be zero when a and b are not?

When a is perpendicular to b, axb is zero. C can be zero any time a and b are nonzero but perpendicular.

30 Can you assign a direction to C? If so, what direction would c be for the vectors a and b as shown?

Since A is roughly 45 degrees and B is 315 degrees roughly, the dot product is close to zero. I am not sure if C is just a number or has an angle. I think it doesn't?