Rotation matrices

5

If I have a rotation matrix to transform a point around the XY plane, how can I tell the angle of rotation ? How do I know whether
the rotation is positive or negative? If given a rotation matrix, in the textbook, how can I determine the angle of rotation and its
direction?

Reply 1

mqb2766

21

Original post

by Skybird

If I have a rotation matrix to transform a point around the XY plane, how can I tell the angle of rotation ? How do I know whether
the rotation is positive or negative? If given a rotation matrix, in the textbook, how can I determine the angle of rotation and its
direction?

The rotation matrix is composed of cos(theta) and sin(theta), so just do arc*** on one of the elements and a bit of reasoning about which quadrant it must be in.

Reply 2

Skybird

OP

5

I have deduced that in the question I am attempting to answer the angle of rotation is 30 degrees and the point lies in the second quadrant. But is the answer 180 degrees - 30 degrees or is it 90 degrees + 30 degrees ? How do I tell?

Reply 3

mqb2766

21

Original post

by Skybird

I have deduced that in the question I am attempting to answer the angle of rotation is 30 degrees and the point lies in the second quadrant. But is the answer 180 degrees - 30 degrees or is it 90 degrees + 30 degrees ? How do I tell?

In the first column you have cos(theta) and sin(theta). Both will be positive in quadrant 1 (A), -/+ in quadrant 2 (S), -/- in quadrant 3 (T) and +/- in quadrant 4 (C). Its just cast or the unit hypotenuse right triangle with base (x-axis) cos(theta) and opposite (y-axis) sin(theta). Then theta is measured from the positive x-axis, so positive anticlockwise and negative clockwise. You should have come across this?

If youre unsure, post an actual question and what you think/youve tried.

Reply 4

Skybird

OP

5

In simpler terms I want to find the direction of rotation from a matrix.

I am trying to answer parts (iii) and (iv) of question 3 above.

Reply 5

Skybird

OP

5

Sorry about the size of the picture.

Reply 6

mqb2766

21

Original post

by Skybird

In simpler terms I want to find the direction of rotation from a matrix.

I am trying to answer parts (iii) and (iv) of question 3 above.

It would help to say if youve covered/googled cast or the unit hypotenuse right triangle? Can you sketch the (cast/right triangle) angle/quadrants?

Both cos(theta) and sin(theta) are -1/sqrt(2) (first column), so their signs implies quadrant 3 (T) and both adjacent and opposite sides are length 1/sqrt(2) so thats an isosceles right triangle with angle ... Also you can measure the angle anticlockwise or clockwise from the positive x-axis if you want a positive or negative rotation, respectively.

Both (iii) and (iv) are pretty much a write down (as they correspond to well known angles/right triangles) with a simple sketch to get the cast part right

Reply 7

tonyiptony

11

The easiest way to tackle the direction problem is to circumvent it by picking a direction (almost always anticlockwise - that's the convention) and stick to it.

Also, by convention, the angle in a rotation matrix is measured anticlockwise.

So you can simply describe a pi/2 (or 90 degree) clockwise rotation as a 3pi/2 (or 270 degree) anticlockwise rotation. They mean the same thing.

(edited 1 year ago)

Reply 8

Muttley79

20

Original post

by Skybird

If I have a rotation matrix to transform a point around the XY plane, how can I tell the angle of rotation ? How do I know whether
the rotation is positive or negative? If given a rotation matrix, in the textbook, how can I determine the angle of rotation and its
direction?

Use base vectors (1, 0) goes to the first column and (0,1) to the second. A sketch should make it clear.

For rotation +90 is the same as - 270

Rotation matrices

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