Rotation Matrices

Part 1: Trigonometric Background

1. Let (x,y) denote the terminal point of the vector v in the plane. Suppose we rotate this vector through an angle and let (x_r, y_r) denote the new terminal point v_r after the rotation. Our first objective is to calculate the new coordinates in terms of the old ones. If b = |v|, then x = b cos() and y = b sin(). Similar formulas give us x_r and y_r in terms of the angle + . Enter the formulas for x_r and y_r, and use the expand function to rewrite these formulas in terms of the angles and . Then identify occurrences of x and y in these expressions, and rewrite the formulas to give x_r and y_r in terms of x, y, and .
2. Rewrite the transformation (rotation through an angle ) as a matrix-vector equation, v_r = Av. That is, give the matrix A.

---

modules at math.duke.edu