Linear algebra with dot product?

[SpiffyEh]

Homework Statement

Let A(theta)x = b for each theta in S. Calculate,
(x dot b)/(|x||b|)

A =
[cos(theta) -sin(theta)
sin(theta) cos(theta)]

How is this related to theta?
Recall that x dot y and |x| are the standard dot-product and magnitude, respectively, from vector-calculus. These operations hold for vectors in Rn
but now have the following definitions, x dot y = x(transpose)y and |x =sqrt(x(transpose)*x)

Homework Equations

The Attempt at a Solution

I'm not sure how to do this because i don't know what x is or what b is, so I'm confused.

 

[Mark44]

x is a vector in R², and so is b. The matrix A is a rotation matrix, where the parameter theta indicates how much rotation.

 

[SpiffyEh]

I don't understand how I would go about calculating it without knowing anything besides that

 

[Mark44]

Pick a value for theta, then pick a few values for x. Now calculate Ax. That should give you an idea of what A does, and how A is related to theta.

 

[SpiffyEh]

don't I need to actually calculate the dot product equation though? I can't just pick something for x and b and actually have it be correct to the equation.

 

[hunt_mat]

The equation is:$$A(\theta )x=b$$. Then $$x=A^{-1}(\theta )b$$, You can calculate what the inverse of A is. Denote $$\mathbf{b}=(b_{1},b_{2})$$, you can then calculate x in terms of theta and b, from there compute your dot product.

 

[HallsofIvy]

Nice to know: if $A(\theta)$ is 'rotation about a given axis by angle $\theta$' then the inverse rotation is just the rotation about the same axis by angle $-\theta$.

That is, $A(\theta)^{-1}= A(-\theta)$.