Question

Which of the following is the image of D after a rotation of 90° about the origin?

(-2, 1)
(2, 1)
(-1, -2)

Answer 1

The answer is (2,1)

Explanation:

Given a vector in the plane, we can describe the rotation as a function :

X is the vector

T is the function that represents the rotation

`T(X)=AX`

Where A is the rotation matrix

The rotation matrix about the origin in counterclockwise sense for an angle ''a'' is :

`A=\left[\begin{array}{cc}cos(a)&-sin(a)\\sin(a)&cos(a)\end{array}\right]`

If we replace ''a'' by 90 ⇒

`\left[\begin{array}{cc}cos(90)&-sin(90)\\sin(90)&cos(90)\\\end{array}\right]`

`A=\left[\begin{array}{cc}0&-1\\1&0\\\end{array}\right]`

If we want to rotate the vector D = (1,-2) :

`T(D)=AD=\left[\begin{array}{cc}0&-1\\1&0\\\end{array}\right]\left[\begin{array}{c}1&-2\end{array}\right]=\left[\begin{array}{c}2&1\end{array}\right]`

The rotation of the vector (1,-2) about the origin for an angle of 90 and in counterclockwise sense is (2,1)

Answer 2

D would be located at (2,1)