Vector Dot Product and Angle Between Two Vectors
Vector Dot Product and Angle Between Two Vectors
Many physics and engineering quantities — work done by a force, projection of one direction onto another, or the angle between two structural members — reduce to the dot product of two vectors. Because the dot product also encodes the cosine of the angle between the vectors, it is a direct route from raw components to a physically meaningful angle.This calculation is a building block used everywhere from computing the component of gravity along an inclined surface to testing whether two vectors are perpendicular (dot product = 0) or nearly parallel (dot product close to the product of the magnitudes).
The dot product is a·b = a_1b_1+a_2b_2+a_3b_3, and the angle between the vectors is θ = arccos(a·b/(|a||b|)). where a_1v, a_2v, a_3v are the components of vector a, b_1v, b_2v, b_3v are the components of vector b, mag_a and mag_b are their magnitudes, and theta_v is the angle between them.
Inputs
a_1v := 3=3
a_2v := 4=4
a_3v := 0=0
b_1v := 1=1
b_2v := 2=2
b_3v := 2=2
Multiply matching components together and sum them — this is the definition of the dot product.
dot_v := a_1v·b_1v + a_2v·b_2v + a_3v·b_3v=11
The magnitude of a vector is the square root of the sum of its squared components, i.e. its length by the Pythagorean theorem in 3D.
mag_a := sqrt(a_1v^2+a_2v^2+a_3v^2)=5
Compute the magnitude of the second vector the same way.
mag_b := sqrt(b_1v^2+b_2v^2+b_3v^2)=3
The dot product equals |a||b|cos θ, so dividing it by the product of the magnitudes and taking the inverse cosine recovers the angle between the two vectors.
theta_v := acos(dot_v/(mag_a·mag_b))=0.75 rad
Results
The two vectors here point in noticeably different directions, so the computed angle is a moderate value well short of parallel (near 0) or perpendicular (π/2). A result near zero would mean the vectors are nearly aligned, while a result near π/2 radians would mean they are nearly perpendicular — both useful sanity checks when the vectors represent physical directions like force or velocity.
Assumptions & limitations
- Vectors given in a common Cartesian coordinate system.
- Angle result is in radians, between 0 and π.
- Neither vector is the zero vector (division by |a||b| would be undefined).