Angle Between Two Vectors Calculator

An online angle between two vectors calculator allows you to find the angle, magnitude, and dot product between the two vectors. It does not matter whether the vector data is 2D or 3D, our calculator works well in all aspects. So, keep reading to learn how to use formulas and some examples to find angle between two vectors.

What is Angle Between Two Vectors?

In mathematics, the angle between two vectors is defined as the smallest angle through which one vector must rotate to align with the other. Vector quantities have both magnitude and direction, and they can be represented in two-dimensional or three-dimensional spaces.

Angle Between Two Vectors Formula:

There are different formulas used by the angle between two vectors calculator, depending on the vector representation:

Find Angle between Two 2D Vectors:

• Vectors represented by coordinates:

Let vectors be \( \mathbf{m} = [x_m, y_m] \) and \( \mathbf{n} = [x_n, y_n] \)

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 \( \mathbf{p} \) with points M = \([x_m, y_m]\) and N = \([x_n, y_n]\):

\( \mathbf{p} = [x_n - x_m, y_n - y_m] \)

For vector \( \mathbf{q} \) with points C = \([x_c, y_c]\) and D = \([x_d, y_d]\):

\( \mathbf{q} = [x_d - x_c, y_d - y_c] \)

Then, substituting into the angle formula:

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

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

Find Angle between Two 3D Vectors:

To calculate the angle between two 3D vectors, the calculator uses the following approach:

• Vectors represented by coordinates:

Let two 3D vectors be:

\( \mathbf{m} = [x_m, y_m, z_m], \quad \mathbf{n} = [x_n, y_n, z_n] \)

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 \( \mathbf{a} \) with points M = \([x_m, y_m, z_m]\) and N = \([x_n, y_n, z_n]\):

\( \mathbf{a} = [x_n - x_m, y_n - y_m, z_n - z_m] \)

For vector \( \mathbf{b} \) with points O = \([x_o, y_o, z_o]\) and P = \([x_p, y_p, z_p]\):

\( \mathbf{b} = [x_p - x_o, y_p - y_o, z_p - z_o] \)

The formula analogical to the 2D case is:

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

You can determine the angle either using coordinates directly or using terminal points with an angle between two vectors calculator.

How to Find the Angle between Two Vectors:

The angle between two vectors can be found using different formulas derived from the dot product:

• Dot Product Formula:

The dot product of two vectors is the product of their magnitudes multiplied by the cosine of the angle between them (\(\theta\)):

\( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \cos(\theta) \)

• Solving for the angle:

Divide both sides by the product of the magnitudes of the vectors:

\( \cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| \cdot |\mathbf{b}|} \)

Then, take the inverse cosine to find the angle:

\( \theta = \cos^{-1} \left( \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| \cdot |\mathbf{b}|} \right) \)

• Note: The magnitude of a vector is the square root of the sum of the squares of its components:

\( |\mathbf{a}| = \sqrt{x_a^2 + y_a^2 + z_a^2} \)

Vector Magnitude:

Vector in 2D space:

\(|\mathbf{v}| = \sqrt{x^2 + y^2}\)

Vector in 3D space:

\(|\mathbf{v}| = \sqrt{x^2 + y^2 + z^2}\)

• The angle between two vectors calculator uses the dot product formula combined with magnitudes:

2D space: For vectors \(\mathbf{m} = [x_m, y_m]\) and \(\mathbf{n} = [x_n, y_n]\):

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

3D space: For vectors \(\mathbf{m} = [x_m, y_m, z_m]\) and \(\mathbf{n} = [x_n, y_n, z_n]\):

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

These formulas are applied by angle between vectors calculators for both 2D and 3D vectors.

Alternatively, you can use a free online Cosine Calculator to compute the cosine value of the angle in degrees or radians.

Example: Find the Angle Between Two 3D Vectors

Given vectors:

\(\mathbf{A} = \{4, 6, 8\}\)

\(\mathbf{B} = \{3, 2, 5\}\)

Step 1: Compute the Dot Product \( \mathbf{A} \cdot \mathbf{B} \)

\(\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z\)

\(\mathbf{A} \cdot \mathbf{B} = (4 \cdot 3) + (6 \cdot 2) + (8 \cdot 5)\)

\(\mathbf{A} \cdot \mathbf{B} = 12 + 12 + 40\)

\(\mathbf{A} \cdot \mathbf{B} = 64\)

Step 2: Compute the Magnitude of Vector \( \mathbf{A} \)

\(|\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2} = \sqrt{4^2 + 6^2 + 8^2} = \sqrt{16 + 36 + 64} = 10.77033\)

Step 3: Compute the Magnitude of Vector \( \mathbf{B} \)

\(|\mathbf{B}| = \sqrt{B_x^2 + B_y^2 + B_z^2} = \sqrt{3^2 + 2^2 + 5^2} = \sqrt{9 + 4 + 25} = 6.16441\)

Step 4: Compute the Angle Between Vectors

\(\cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| \cdot |\mathbf{B}|} = \frac{64}{10.77033 \cdot 6.16441} = \frac{64}{66.39277} = 0.9639604\)

\(\theta = \cos^{-1}(0.9639604) \approx 15.429^\circ\)

For exact results, you can also input these values into an angle between vectors calculator.

How the Angle Between Two Vectors Calculator Works

The angle between two vectors calculator finds the angle θ separating vectors \(\mathbf{A}\) and \(\mathbf{B}\) in 2D or 3D space using the following steps:

Input:

• Select the vector dimension: 2D or 3D.
• Choose the vector representation: either by Coordinates or Terminal Points from the drop-down list.
• Enter all given values in the corresponding fields.
• Click the “Calculate” button to see the results.

Output:

• The calculator provides step-by-step calculations for the dot product, vector magnitudes, and the angle between the vectors.
• You can recalculate multiple 2D or 3D vectors as needed by clicking the "Recalculate" button.

FAQ

Define the angle between two antiparallel vectors?

Antiparallel vectors are parallel vectors pointing in opposite directions. The angle between them is 180°.

What is the magnitude of a unit vector?

The magnitude of a unit vector is 1 by definition. If you want to use the unit vector in any direction, its magnitude ensures proper scaling in that direction.

Is the angle a vector quantity?

Angles are considered dimensionless quantities with direction and magnitude. Clockwise and counterclockwise rotations can define their orientation, so in certain contexts, the angle behaves like a vector quantity.

Conclusion

Use this angle between two vectors calculator to quickly determine the angle between vector components. Understanding vector angles is important to describe the angular difference between physical quantities with direction. This free online calculator efficiently provides the dot product, vector magnitudes, and the resulting angle with high accuracy.

References

From Wikipedia: Algebraic and geometric definitions, Scalar projection, Equivalence of definitions, Application to the law of cosines, Vector algebra relations, Magnitudes.

From WikiHow: Finding the Angle Between Two Vectors, Angle formula, Law of Cosines applied to vectors, Read more.

From Krista King Math: Angle Between Two Vectors, Formula derivation, How to calculate angles, Finding angles in 3D.