To rotate a vector by 90 degrees, you can use the concept of representing the vector as a complex number and multiplying it by the imaginary unit i. This method, as highlighted in the provided reference, directly results in a 90-degree counterclockwise rotation.
Understanding 90-Degree Vector Rotation
A 2D vector can be thought of as a point or a complex number in the complex plane. A vector with coordinates (a, b) can be represented as the complex number a + bi. The geometric interpretation of multiplying a complex number by the imaginary unit i is a rotation by 90 degrees counterclockwise around the origin (0, 0).
According to the reference, "A 90 degree rotation is the result of multiplying by the complex number i, or 1 * cis(90)."
The Method Using Complex Numbers
Here's how to rotate a vector (a, b) by 90 degrees counterclockwise using this method:
- Represent the vector as a complex number: A vector (a, b) corresponds to the complex number a + bi.
- Multiply the complex number by i: Perform the multiplication (a + bi) * i.
- Simplify the result:
(a + bi) * i = (a * i) + (bi * i)
= ai + bi²
Since i² = -1, this simplifies to:
= ai - b
Rearranging the terms to the standard form (real + imaginary):
= -b + ai - Convert the resulting complex number back to a vector: The complex number -b + ai corresponds to the vector (-b, a).
Therefore, rotating the vector (a, b) by 90 degrees counterclockwise results in the new vector (-b, a).
Example: Rotating the Vector (3, 4)
Let's apply this method to a specific example, as mentioned in the reference. Consider the vector (3, 4).
- This vector corresponds to the complex number 3 + 4i.
- To rotate it by 90 degrees, we multiply by i:
(3 + 4i) * i - The calculation is:
(3 + 4i) * i = (3 * i) + (4i * i)
= 3i + 4i²
= 3i + 4(-1)
= -4 + 3i
The reference states the result would be "-4 + 3i".
- Converting the resulting complex number -4 + 3i back to a vector gives (-4, 3). The reference confirms this, stating the result "if translated to a vector would simply be a new vector with the values of (-4, 3)."
So, rotating the vector (3, 4) by 90 degrees counterclockwise results in the vector (-4, 3).
General Formula for 90-Degree Counterclockwise Rotation
Based on the complex number multiplication method:
| Original Vector (a, b) | Rotated Vector (90° Counterclockwise) |
|---|---|
| (a, b) | (-b, a) |
| (3, 4) | (-4, 3) |
Rotating Clockwise
While the primary method involves counterclockwise rotation, rotating a vector (a, b) by 90 degrees clockwise is equivalent to multiplying the complex number (a + bi) by -i.
(a + bi) * (-i) = -ai - bi² = -ai - b(-1) = -ai + b = b - ai.
This corresponds to the vector (b, -a).
In summary, the standard 90-degree rotation typically refers to the counterclockwise rotation, which transforms a vector (a, b) into (-b, a).
