Linear algebra - Rotations in R3

[SpiffyEh]

Homework Statement

Given,

R_1(x) =
[ 1 0 0
0 cos(x) -sin(x)
0 sin(x) cos(x)]

R_2(x) =
[cos(x) 0 -sin(x)
0 1 0
sin(x) 0 cos(x)]

R_3(x) =
[cos(x) -sin(x) 0
sin(x) cos(x) 0
0 0 1]

Describe the transformations defined by each of these matrices on vectors in R3.

Homework Equations

The Attempt at a Solution

I have no idea how to describe these or even go about figuring it out. Can someone explain to me how to do one of them or at least guide me through the process. Then I can attempt the rest please?

 

[Mark44]

You might start by applying these transformations on some vector to see what happens. To decrease confusion, I would use t as the parameter in all three transformations, instead of x.

For R_1, the x-coordinate of an input vector will be unchanged, so it seems to me that the rotation is entirely in the y-z plane. Test this by choosing a value of t such as pi/2, and seeing what happens to <1, 1, 1>

 

[SpiffyEh]

You might start by applying these transformations on some vector to see what happens. To decrease confusion, I would use t as the parameter in all three transformations, instead of x.

For R_1, the x-coordinate of an input vector will be unchanged, so it seems to me that the rotation is entirely in the y-z plane. Test this by choosing a value of t such as pi/2, and seeing what happens to <1, 1, 1>

doesn't it just change the y component? If i multiply i get <1, -1, 1>

 

[Mark44]

No, the first transformation actually changes the y and z components, and leaves the x component unchanged. You need a vector with different components to see the change.

 

[SpiffyEh]

oh ok, when i tested it with <1,2,3> i ended up with <1,-3,2> so it seems like its just flipping the y and z components and negating the y. I'm not exactly sure what this means in the yz-plane, I don't think that would be enough to say

 

[Mark44]

What it means is that R_1(x) rotates the projection of x in the y-z plane.

 

[SpiffyEh]

What it means is that R_1(x) rotates the projection of x in the y-z plane.

Can i say that it is rotated clockwise? Because if i graph it seems to go clockwise

 

[Mark44]

Yes, I believe that is correct, as long as it's clear what clockwise means. In three dimensions "clockwise" doesn't have much meaning, but if you state the the projection of x in the y-z plane is being rotated clockwise, then that's clear.

 

[SpiffyEh]

ok, that makes sense. I understand how to go about doing the other two now. Thank you so much for all the help. It actually makes sense now