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

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