Angle Between Two Vectors Calculator

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This online Angle Between Two Vectors Calculator finds the angle between two vectors defined in 2D or 3D Cartesian coordinate system. You can paste input vector components copied from a spreadsheet or csv file, or enter manually using comma, space, or enter as delimiters. The resulting angle can be represented in degrees or radians.

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Precision: decimal places

Vector A

Vector B

Degrees   Radians

Angle:

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What is a Vector

Vectors are fundamental mathematical objects that play an important role in various scientific disciplines, including physics and engineering. They provide a concise and efficient way of representing quantities that have both magnitude and direction.

In mathematics, a vector is an object consisting of two main components: magnitude (or length) and direction in space. In diagrams and graphs, vectors are usually depicted with arrows and denoted by bold letters or letters with arrows, such as \(\textbf{V}\) or \(\overrightarrow{V}\).

Any vector can be uniquely defined by its projections on the coordinate axes in the space in which it acts. In other words, a vector can be algebraically defined by a set of numbers that can be considered as its components. For example, in a three-dimensional Cartesian coordinate system, any vector can be represented as follows:

$$\overrightarrow{V} = \{ V_x, V_y, V_z \}.$$

Angle Between Two Vectors

The angle between two vectors is the measure of the rotation required to align one vector with the other. It is expressed in degrees or radians and can be calculated using various formulas, depending on the vector representation and the coordinate system being used.

Let’s consider two vectors, \(\overrightarrow{A}\) and \(\overrightarrow{B}\), in 2D space. These vectors can be represented using their components, \(\overrightarrow{A} = \{ A_x, A_y \}\) and \(\overrightarrow{B} = \{B_x, B_y\}\), respectively. The angle between these two vectors can be determined using the dot product formula:

$$\theta = arccos \left(\frac{\overrightarrow{A} \cdot \overrightarrow{B}}{|\overrightarrow{A}| |
\overrightarrow{B}|}\right),$$

where
• \(\theta\) is the angle between the vectors \(\overrightarrow{A}\) and \(\overrightarrow{B}\) in the plane containing them,
• \(arccos\) is the principal value of the arccosine function, therefore \( 0 \leq \theta \leq \pi \),
• \(\overrightarrow{A} \cdot \overrightarrow{B}\) is the dot product (scalar product) of the two vectors,
• \(|\overrightarrow{A}|\) and \(|\overrightarrow{B}|\) represent the magnitudes of the vectors.

As is well known, the following expressions are valid for the magnitudes of vectors and their scalar product:

$$|\overrightarrow{A}| = \sqrt{A_x A_x + A_y A_y} \ , \ |\overrightarrow{B}| = \sqrt{B_x B_x + B_y B_y} \ ,$$

$$\overrightarrow{A} \cdot \overrightarrow{B} = A_x B_x + A_y B_y \ .$$

As you can see from the above formula for the angle \(\theta\), it does not depend on the dimension of the space and thus is also valid for 3D vectors. It is only necessary to add the third component to the formulas defining the magnitudes of the vectors and their scalar product:

$$|\overrightarrow{A}| = \sqrt{A_x A_x + A_y A_y + A_z A_z} \ , \ |\overrightarrow{B}| = \sqrt{B_x B_x + B_y B_y + B_z B_z} \ ,$$

$$\overrightarrow{A} \cdot \overrightarrow{B} = A_x B_x + A_y B_y + A_z B_z \ .$$

The above formulas are used in our Angle Between Two Vectors Calculator to calculate the angle between vectors. Depending on whether the vectors are 2D or 3D vectors, either two or three components of these vectors are entered into the input fields of the calculator.