Order of operations in rotation matrix notation.

I'm trying to convert this equation to C# but I'm not a mathematician and I find math notation ambiguous:

See the first matrix in this article:

http://mathworld.wolfram.com/RotationMatrix.html

which has:

$$R'_\theta=\begin{bmatrix}\cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{bmatrix}$$

That $-\sin\theta$, is that (negate sin) × angle, or negate (sin × angle)?

In math, unlike programming, there's not much use of parentheses in math equations, so what in what order should calculations be done?

• the 0 with the line is the greek letter theta and you can type it by typing \$\theta\$ – mv3 Mar 12 '13 at 18:17
• It's sine of the angle theta, not "sine times theta" (whatever that would mean). In most programming languages you'd write it as "-sin(theta)". – Hans Lundmark Mar 12 '13 at 18:19
• unfortunately there's no ambiguity here. in math, like programming, we use lots of parentheses to avoid ambiguity. – Tyler Mar 12 '13 at 18:21
• Even if $\sin$ were some quantity that you were multiplying by $\theta$ (which I'll admit is what it looks like if you don't know), the order of operations wouldn't make any difference here. Negation is multiplication by $-1$, and multiplication is commutative, meaning you can multiply a list of numbers in any order and get the same result. – mdp Mar 12 '13 at 18:30
• Should you need to post here again, this article has a summary of how to format mathematics, including that o with a line through it. – MJD Mar 12 '13 at 18:43

$\theta$ is the greek letter theta. In this context, $\theta$ represents the angle of rotation.

$\sin$ and $\cos$ are trigonometric functions. You can read about them here. In your computations, these should be evaluated first.

If you have a point on a plane represented by the tuple $(x,y)$, mathematicians refer to this as a vector in $\mathbb{R}^2$. ($\mathbb{R}^2$ is the plane of real numbers.) Matrices like $R_\theta$ are transformations which we apply to vectors. If we let $$R=\left[\begin{array}{cc}r_{11}&r_{12}\\r_{21}&r_{22}\end{array}\right]$$ and multiply this with the vector $v=\left[\begin{array}{c}v_1\\v_2\end{array}\right]$, we have $$Rv=\left[\begin{array}{cc}r_{11}&r_{12}\\r_{21}&r_{22}\end{array}\right]\left[\begin{array}{c}v_1\\v_2\end{array}\right]=\left[\begin{array}{c}r_{11}v_1+r_{12}v_2\\r_{21}v_1+r_{22}v_2\end{array}\right].$$ So in the case of $R_\theta$, you have $r_{11}=\cos\theta$, $r_{12}=-\sin\theta$, etc. Compute these before you do the matrix multiplication, and you can use the above formula.

Of course, when you get to vectors in $\mathbb{R}_3$ (associated with $3\times 3$ matrices) you will need to use a different formula. $\text{C}\#$ probably has some built in commands to deal with all this. I would recommend reading up on the documentation if you want to avoid doing the math yourself.

• This is well-paced, comprehensive and clear, thank you. In C# there's only the most basic math building blocks and community projects tend to make up the difference. For technical reasons I can't use them, but I'm looking to improve my mathematics. – Luke Puplett Mar 12 '13 at 19:03
• @LukePuplett Well, you've come to the right place, then. :) Welcome to MSE. – Alexander Gruber Mar 12 '13 at 19:05

The expression is

$$-\sin\theta = -(\sin\theta)$$

By "the equation" I assume you mean the rotational transformation in $\mathbb{R}^2$. In general you can expand the multiplication to get a vector valued function $\vec F$, so $\vec F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, and it is defined by

$$\vec F(x,y) = (x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta)$$

Which looks pretty much like a tuple. I'm sure there are math libraries for C# out there that have $\sin$ and $\cos$, so you can just use those.

• System.Math.sin and System.Math.cos in C#. – Emily Mar 12 '13 at 18:24