You can find a unit vector given an angle by using the trigonometric functions cosine and sine based on the angle measured from the positive x-axis.
A unit vector is a vector with a magnitude (or length) of 1. It's used purely to indicate a direction. When you have a direction specified by an angle in a 2D coordinate system, you can directly determine the components of the unit vector pointing in that direction.
The Formula
The key is to measure the angle from the positive x-axis, rotating counterclockwise. Let this angle be θ (theta).
According to the principles of trigonometry and the unit circle, the coordinates of the point where the terminal side of the angle θ intersects the unit circle are (cos θ, sin θ). Since a unit vector starts at the origin (0,0) and ends at this point on the unit circle, these coordinates are precisely the components of the unit vector.
- x-component: cos θ
- y-component: sin θ
So, the unit vector u is given by:
u = (cos θ, sin θ)
As mentioned in the reference, this process goes "all the way from the X [axis] to the unit vector itself," highlighting the importance of measuring the angle relative to the x-axis in the coordinate system to find the vector's components.
Practical Steps
- Identify the Angle: Determine the angle θ that the desired unit vector makes with the positive x-axis, measured counterclockwise.
- Calculate Cosine: Find the cosine of the angle θ. This will be the x-component of the unit vector.
- Calculate Sine: Find the sine of the angle θ. This will be the y-component of the unit vector.
- Form the Vector: Combine the calculated cosine and sine values to form the unit vector (cos θ, sin θ).
Example: Finding the Unit Vector at 45 Degrees
Let's find the unit vector at an angle of θ = 45 degrees from the positive x-axis, as discussed in the reference where theta is 45 degrees within a quadrant.
- Angle: θ = 45°
- x-component (cos 45°): cos(45°) = √2/2
- y-component (sin 45°): sin(45°) = √2/2
Therefore, the unit vector at 45 degrees is (√2/2, √2/2).
You can verify this is a unit vector by calculating its magnitude:
Magnitude = √[(√2/2)² + (√2/2)²] = √[(2/4) + (2/4)] = √[1/2 + 1/2] = √1 = 1.
This confirms it is indeed a unit vector.
Using this method, you can find the unit vector for any given angle, regardless of the quadrant, by simply evaluating the cosine and sine of that angle.
