Question

Point P(3, 9) in the coordinate plane is rotated 90 degrees counterclockwise about the origin. What are the coordinates of its image?

Please show all work and not just the answers.

Answer 1

P'(-9, 3)

Explanation:

The transformation for rotation counterclockwise about the origin by some angle α will be ...

(x, y) ⇒ (x·cos(α) -y·sin(α), x·sin(α) +y·cos(α))

When the angle is α = 90°, this reduces to ...

(x, y) ⇒ (-y, x)

You have (x, y) = (3, 9), so the image point is P'(-9, 3).

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Additional comment

The "work" is applying the right sign and choosing the right coordinate to fill in the values (-y, x). I find it easier to look up the transformations for ±90° and 180°, rather than try to derive them from first principles every time. Your text may have a list:

• 90° CCW or 270° CW: (x, y) ⇒ (-y, x)
• 90° CW or 270° CCW: (x, y) ⇒ (y, -x)
• 180°: (x, y) ⇒ (-x, -y)