Science:Math Exam Resources/Courses/MATH152/April 2011/Question A 15/Solution 1

From UBC Wiki

We could start by finding both of the separate transformation matrices and then multiplying them together which would give us the general transformation matrix T. However since we only need T([1,0]), the transformation of the unit vector in the x-direction, it will suffice to just consider how that single vector is transformed and not deal with matrices at all.

We first rotate the vector [1,0] counter-clockwise by to get a new vector . After doing so, the new vector forms a right triangle with a vector in the x-direction and a vector in the y-direction. We can get its components by using trigonometry. The new vector will have length 1 (since it is a rotation of [1,0] which also has length 1) so its components are

We then must project this vector onto u=[1,2]. Recall that a projection of a vector v onto a vector u is

Using our vector and projecting it onto u which has length , we have that our new vector, , is

Therefore, the transformation on the vector [1,0] is