# How do I graph a vector [cos x, sin x]?

how do I do the following:

Consider the matrix $\begin{bmatrix} \cos x & -\sin x\\ \sin x & \cos x \end{bmatrix}$ and the vector $\begin{bmatrix} \cos y\\ \sin y \end{bmatrix}$. Show that the product of the $2\times 2$ matrix with $2\times 1$ matrix $\begin{bmatrix} \cos(x+y)\\ \sin(x+y) \end{bmatrix}$ graphically, by sketching.

I can show this using trig identities, but I am not sure how to graph the $2\times 1$ trig vector in first place, let alone explain the rest of the question graphically. All the help very much appreciated!

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[cos(y),sin(y)] is the unique point on the unit circle with angle $y$ from the origin. Your $2x2$ matrix rotates the plane about the origin by angle $x$, which is why the result is a point on the same circle with angle $x+y$.

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Thank you very much! –  LinAlgStudent Jan 9 '11 at 0:51
Every point $(x,y)$ can be thought of as a $2 \times 1$ matrix.
If $(x,y)$ is a point in the 2D plane, then for a $2\times 2$ matrix $A$, the product of $A$ and $[x,y]^{T}$ gives another 2D point.
In this case, you $A$ is a rotation matrix, and the $2 \times 1$ matrix you have corresponds to a point on the unit circle.