Really fundamental cross/dot product questions

[CookieSalesman]

So I've been working on physics homework and we have some vector/dot product questions.
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($\theta$).
Dot products seem to be abCosine($\theta$).

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 $\theta$ 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 parallel 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$\bullet$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?

 

[Quesadilla]

You need to read up on the differences between scalars and vectors. You also need to read about cross products.

A scalar is just a (real) number.

A vector has both magnitude and direction. You should think of it as an arrow in the coordinate system. It has a certain length (its magnitude) and it points somewhere (its direction).

The dot product takes two vectors and produces a scalar.

The cross product takes two vectors and produces a new vector.

The formula you have for the cross product is only for its magnitude. $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin \theta$, where $\theta$ indeed is the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$.

Your book should also tell you something about how you determine the direction of the vector produced by the cross product.

 

[Fredrik]

Since this is the homework forum, I can only give you hints, not complete answers.

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).

The only part of that that's relevant here is that they're not parallel, and not 0.

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

Cross products seem to be |a||b|sine($\theta$).
Dot products seem to be abCosine($\theta$).

The dot product $x\cdot y$ is $|x||y|\cos\theta$ where $\theta$ is the angle between the vectors, but the cross product $x\times y$ is a vector with magnitude $|x||y|\sin\theta$.

Note that the definitions make it clear that $x\cdot y$ is a number and $x\times y$ is a vector:
\begin{align}
&x\cdot y=x_1y_1+x_2y_2+x_3y_3\\
&x\times y=(x_2y_3-x_3y_2,x_3y_1-x_1y_3,x_1y_2-x_2y_3)
\end{align}

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 $\theta$ 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.

That angle only contributes to the magnitude, not the direction. The direction is always perpendicular to the plane that contains x and y. A lot of people use a "right-hand rule" to remember which of the two perpendicular directions it is. http://en.wikipedia.org/wiki/Right-hand_rule

The angle has nothing to do with coordinate axes. There's a unique plane that contains x,y and 0. The angle is measured in that plane.

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.

You have a formula for the magnitude, so I suggest that you use it. Cross products aren't commutative. They're not even associative. Use the definition that I included above, or the right-hand rule, to figure out the direction.

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 parallel to B.

That's a good start, but there are actually two directions that work (two angles).

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.

The point is that the axes are irrelevant, or equivalently, that your choice of basis for $\mathbb R^3$ is irrelevant. This isn't obvious, since the definition I included above can give the impression that the basis matters. The numbers $x_1,x_2,x_3,y_1,y_2,y_3$ all depend on it.

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$\bullet$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?

Scalars don't have angles (except when we're talking about complex scalars, but there's no need to talk about that now). A scalar is a number. A vector in $\mathbb R^3$ is an ordered triple of numbers $(x_1,x_2,x_3)$.