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Angle Between Two Vectors Calculator

Select the vector type, choose the representation method, and enter all coordinates. The calculator will instantly determine the angle, magnitude, and dot product between them.

Vector

3D Dimensions 2D Dimensions

Dimensions(A) Representation:

Initial Point (a)

First vector (A)

Second vector (B)

Dot Product of Vectors (m):

Terminal Point (b)

First vector (A)

Second vector (B)

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An online angle between two vectors calculator allows you to find the angle, magnitude, and dot product between two vectors. It works for both 2D and 3D vectors. Read on to learn how to use formulas and examples to determine the angle between vectors.

What is Angle Between Two Vectors?

In mathematics, the angle between two vectors is the smallest angle through which one vector must rotate to align with the other. Vectors have both magnitude and direction and can be represented in 2D or 3D space.

Angle Between Two Vectors Formula

Different formulas are used depending on how the vectors are represented:

1. 2D Vectors

Vectors represented by coordinates:

Let vectors be m = [x_m, y_m] and n = [x_n, y_n]. Then the angle between them is:

\( \text{Angle} = \cos^{-1} \left[ \frac{x_m x_n + y_m y_n}{\sqrt{x_m^2 + y_m^2} \cdot \sqrt{x_n^2 + y_n^2}} \right] \)

Vectors defined by terminal points:

For vector p with points M = [x_m, y_m] and N = [x_n, y_n]:

p = [x_n − x_m, y_n − y_m]

For vector q with points C = [x_c, y_c] and D = [x_d, y_d]:

q = [x_d − x_c, y_d − y_c]

Then, the angle is:

\[ \text{Angle} = \cos^{-1} \left[ \frac{(x_n - x_m)(x_d - x_c) + (y_n - y_m)(y_d - y_c)}{\sqrt{(x_n - x_m)^2 + (y_n - y_m)^2} \cdot \sqrt{(x_d - x_c)^2 + (y_d - y_c)^2}} \right] \]

An online Arccos Calculator can compute the inverse cosine for any number.

2. 3D Vectors

Vectors represented by coordinates:

Let vectors be m = [x_m, y_m, z_m] and n = [x_n, y_n, z_n]. The angle between them is:

\[ \text{Angle} = \cos^{-1} \left[ \frac{x_m x_n + y_m y_n + z_m z_n}{\sqrt{x_m^2 + y_m^2 + z_m^2} \cdot \sqrt{x_n^2 + y_n^2 + z_n^2}} \right] \]

Vectors defined by terminal points:

For vector a with points M = [x_m, y_m, z_m] and N = [x_n, y_n, z_n]:

a = [x_n − x_m, y_n − y_m, z_n − z_m]

For vector b with points O = [x_o, y_o, z_o] and P = [x_p, y_p, z_p]:

b = [x_p − x_o, y_p − y_o, z_p − z_o]

The angle formula is analogous to the 2D case:

\[ \text{Angle} = \cos^{-1} \left[ \frac{(x_n - x_m)(x_p - x_o) + (y_n - y_m)(y_p - y_o) + (z_n - z_m)(z_p - z_o)}{\sqrt{(x_n - x_m)^2 + (y_n - y_m)^2 + (z_n - z_m)^2} \cdot \sqrt{(x_p - x_o)^2 + (y_p - y_o)^2 + (z_p - z_o)^2}} \right] \]