Angle Between Two Vectors Calculator

Calculate the angle between two vectors by entering the two-dimensional or three-dimensional coordinates for each vector below.

Vector a
Vector b
Vector a
Vector b

Angle Between Vectors (θ):

Degrees:
 
Radians:
 

Steps to Solve

Use the Angle Between Vectors Formula

θ = cos-1((a · b) / (|a| · |b|))

Given the dot-product and magnitude formulas:

a·b = (xa · xb) + (ya · yb) + (za · zb)

|a|= √(xa² + ya² + za²)

|b|= √(xb² + yb² + zb²)

Substitute Values and Solve

Enter vectors a & b above to see the solution here

Learn how we calculated this below


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How to Find the Angle Between Two Vectors

In linear algebra, vectors represent an ordered sequence of numbers having a direction and a magnitude. When there are multiple vectors, it’s sometimes necessary to find the angle between them.

The angle between vectors is defined as the shortest angle between them, which is taken by aligning the initial point. Ultimately this is the angle formed at the intersection of the initial points, or tails of each vector.

Angle Between Two Vectors Formula

There are two distinct formulas to find the angle between two vectors.

Angle Between Two Vectors using Dot Product

The first formula uses the dot product of the two vectors. This is the formula that the calculator above uses.

θ = cos-1((a · b) / (|a| · |b|))

Thus, the angle θ between the two vectors a and b is equal to the inverse cosine of the dot product a · b divided by the magnitude of vector a |a| times the magnitude of vector b |b|.

You can use our vector magnitude calculator to solve for |a| and |b| in this formula.

Angle Between Two Vectors using Cross Product

The second formula uses the cross product of the two vectors, and it’s very similar to the dot product formula.

θ = sin-1((a × b) / (|a| · |b|))

Using this formula, the angle θ between the two vectors a and b is equal to the inverse sine of the cross product a × b divided by the magnitude of vector a |a| times the magnitude of vector b |b|.

You can also use our vector addition and vector subtraction calculators to perform more operations on vectors.