Proof: Coordinate Rotation Around (0,0)

• Maybe_Memorie
In summary, if you have vectors u = \left(\begin{array}{c}1\\0\end{array}\right) v = \left(\begin{array}{c}0\\1\end{array}\right)and you want to rotate them through an angle \theta, you can use the following formulas:rot_\theta(v) = a \cdot rot_\theta (\left(\begin{array}{c}1\\0\end{array}\right) ) + b\cdot rot_\theta ( \left(\begin{array}{
Maybe_Memorie

Homework Statement

Prove that the coordinates of the point (x',y') where the counter-clockwise rotation through the angle @ around (0,0) brings the given point (x,y) are

x' = xcos@ - ysin@
y' = xsin@ + ycos@

Hint: show that for the points (x,y) = (1,0) and (x,y) = (0,1) directly,
and use the fact that the vector (x,y) is equal to the combination
x.(1,0) + y.(0,1)

Homework Equations

For vectors u and v, angle @ between them
u.v = |u||v|Cos@

The Attempt at a Solution

I don't want to be told how to do it, I would prefer if someone would kind of tease the solution out of me, if you know what i mean..

I've included a diagram, showing my interpretation of the question.

I've tried a few different approaches for the question.
I used the fact that tan@ = (m1 - m2)/(1 +m1m2).
I got the slopes of the lines being y/x and y'/x'. When I plugged everything in and rewrote tan as sin/cos, I got the required formulae, but they were both being divided by each other.

I also used the dot product, put this just resulted with a lot of squares which doesn't help.

I don't entirely understand the hint also.

Attachments

• vector.png
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Use the hint that was given.

First draw a rectangle with corners (0,0) and (x,y) (with sides parallel to x- and y-axis) and then draw the rectangle that you get when the whole plane is rotated by an angle @.

Then you can see in your drawing that the new x' and y' are just sums of the height and width of the rectangle multiplied by sines and/or cosines of @

I don't really understand what the hint means.

Okay, I'll try that.

The hint is basically saying that if your starting vectors are either

$$\left(\begin{array}{c}1\\0\end{array}\right)\mbox{ or }\left(\begin{array}{c}0\\1\end{array}\right)$$

then it's pretty easy to figure out what the new coordinates will be if you rotate it through an angle $$\theta$$.

You'll need the sum of angles formulas for cos and sin, $$\cos(\theta_{0}+\theta)$$ and $$\sin(\theta_{0}+\theta)$$.
($$\theta_{0}$$ is the initial trig angle for the vector).

Does this make sense?

It makes sense, but I have no idea where to start.

You original vector is:

$$v = a\cdot \left(\begin{array}{c}1\\0\end{array}\right) + b\cdot \left(\begin{array}{c}0\\1\end{array}\right)$$

so the rotated vector will be:

$$rot_\theta(v) = a \cdot rot_\theta (\left(\begin{array}{c}1\\0\end{array}\right) ) + b\cdot rot_\theta ( \left(\begin{array}{c}0\\1\end{array}\right) )$$

You can first determine c and d in:

$$rot_\theta (\left(\begin{array}{c}1\\0\end{array}\right) ) = c\cdot \left(\begin{array}{c}1\\0\end{array}\right) + d\cdot \left(\begin{array}{c}0\\1\end{array}\right)$$

and e and f in:

$$rot_\theta (\left(\begin{array}{c}0\\1\end{array}\right) ) = e\cdot \left(\begin{array}{c}1\\0\end{array}\right) + f\cdot \left(\begin{array}{c}0\\1\end{array}\right)$$

and then substitute those in your expression for $$rot_\theta(v)$$.

1. What is coordinate rotation?

Coordinate rotation is a mathematical concept that involves changing the orientation of a coordinate system by rotating it around a specific point, usually the origin (0,0). This is often used in geometry and physics to analyze and understand the relationships between points and shapes.

2. How is coordinate rotation performed?

Coordinate rotation can be performed using various methods, but the most common is the use of matrices. A rotation matrix is a 2x2 matrix that represents the rotation of a point around the origin by a given angle. By multiplying the point's coordinates by the rotation matrix, the point is rotated around the origin.

3. What is the purpose of coordinate rotation?

The purpose of coordinate rotation is to simplify mathematical calculations and analysis by changing the orientation of a coordinate system. This can make it easier to visualize and understand the relationships between points and shapes, and can be particularly useful in calculating distances, angles, and areas.

4. What is the difference between clockwise and counterclockwise rotation?

In coordinate rotation, the direction of rotation can be either clockwise or counterclockwise. Clockwise rotation is when the points are rotated in the direction of the hands of a clock, while counterclockwise rotation is in the opposite direction. This can affect the final coordinates of the rotated points.

5. Can coordinate rotation be applied to 3D coordinates?

Yes, coordinate rotation can also be applied to 3D coordinates. In this case, a 3D rotation matrix is used, which involves rotating the point around each of the three axes (x, y, and z). This is commonly used in 3D graphics and computer games to create the illusion of movement and rotation in 3D space.

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