Vector Rotation: Techniques and Formulas

Rotating a vector Here's the problem: Let v denote a vector that has initial position v1 and final position v2.We know these two numbers. Now suppose we rotate v counterclockwise by an angletheta. The result is a new vector, denoted by w, that also has initial position v1 and finalposition v2. Let's say that we rotate vector v by an angle theta. If we do this, our new vector becomesw. The problem we are going to discuss is how to find w. Time for a warmup with a bit of cases. There is a vector v that points in the x direction. V1 comma 0. So, there we have it. V1 and a comma 0. And then when we rotate v, we get anothervector, w. OK, so what's w? This is actually very similar to problems we've solved beforebecause we know this angle, theta, and we know the length of w. The length of w is equalto its starting point, so it's v1. The length of w equals to the length of v, which is v1. Nowwe want to find the components of w. The first component is v1 times the sine of theta, andthe second component is v1 times the cosine of theta. This is it. What if in the second case, let's assume that v is in the y-direction rather than in thex-direction. So v is 0 comma v2. Thus, there is a vector v. We can rotate this vector by an angle θ and obtain a new vectorw. Please find w.