2D Vector Angle Calculator

Vectors are of great importance in our day-to-day life and numerous different fields and are known to be one of the most fundamental tools in mathematics. The direction and the magnitude of a certain object is defined by vectors. Vectors contain useful information regarding an object's direction and magnitude. You can notice that the applications of vectors are critical in both engineering and science, and it is even more important in pivotal fields such as fluid dynamics, electromagnetic theory, and AC circuit analysis.

2D Vector Angle Calculator
Please enter all values to calculate the 2D Vector Angle
2D Vector Angle Calculator Results
Angle (Degrees) = °
2D Vector Angle Formula and Calculations
θ = arccos(Ax × Bx) + (Ay × Ay)/(A2x + A2y) × (B2x + B2y) × 360/π2
θ = arccos( × ) + ( × )/(2 + 2) × (2 + 2) × 360/π2
θ = arccos( × ) + ( × )/( + ) × ( + ) × 360/π2
θ = arccos + / × × 360/π2
θ = arccos/ × 360/π2
θ = arccos/ × 360/π2
θ = arccos() × 360/π2
θ = × 360/2
θ = /
θ =
2D Vector Angle Calculator Input Values
Vector 1 x-axis (Ax) =
Vector 1 y-axis (Ay) =
Vector 2 x-axis (Bx) =
Vector 2 y-axis (By) =

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2D Vector Angle Calculator. This image shows 2D Vector Angle formula with associated calculations used by the 2D Vector Angle Calculator

2D Vector Angle Calculations Explained

Performing calculations related to vectors in these fields is quite difficult. For that, it is easier to use a calculator and make math calculations more efficient. Having such a calculator that calculates the angle between two vectors without taking much time is nothing less than a boon. This is the reason why we developed the "2D Vector Angle Calculator" to you to provide ease of vector calculations.

Before defining the different aspects of the 2D Vector Angle Calculator, let us revise some other quantities related to it.

Basic Trigonometry

Trigonometry deals with the study of triangles and its different properties such as angles and lengths. Off all the triangles that we study, a right-angles triangle is the most special one. There are three parts of a right-angled triangle known as perpendicular, base and hypotenuse. Moreover, there are three major functions in trigonometry known as the cosine, sine, and tangent. We define these functions as:

sin θ = perpendicular/hypotenuse
cos θ = base/hypotenuse
Tan θ =perpendicular/base

Understanding Vectors

A vector may be defined as an object that constitutes both the magnitude and direction. A vector can be geometrically represented as a line segment whose length defines the magnitude of the vector and an arrow on the line segment that defines the direction of the vector. We generally represent the direction of a vector from its tail to head (an arrow is represented on the head of the vector).

There are three major operations related to vectors known as vector addition, vector multiplication and vector subtraction.

Properties of a Vector

  • Two vectors are said to be equal if and only if they have the same magnitude and direction.
  • A negative vector can be represented just by changing its direction with respect to the reference positive direction.
  • Subtracting two vectors has the same meaning as that of adding two vectors in which one has an opposite direction.
  • The resultant of two vectors is a single vector whose effect is the same as the two vectors acting together individually.

The Angle Between Two Vectors in 2-D

The angle between the two line segments that individually form a vector is known as the angle between those two vectors. When we are talking about vectors in 2-D, we mean to consider the x-axis and y-axis only. Let us understand this with the help of an example.

Let us take two vectors namely "a" and "b". The angle between these two vectors can be represented by the above formula:

cos θ = a.b / |a| |b|


  • cos θ is the angle between the two vectors
  • a.b in the numerator is the scalar product of two vectors
  • the denominator i.e., |a| |b| is the product of the magnitude of the two vectors

The |a| in the denominator defines modular functions which means that only the magnitude of the vector is taken. If you want to find out the length (magnitude) of the vector, you need to add the square of respective coefficients of the vectors and then take the square root of the resultant answer. The final output will be the magnitude of the vectors. After solving the scalar product (also known as the dot product of vectors) and the length of the vector, you can easily solve the fraction. Now you must solve the cosine function in the left-hand side of the equation. For that, you need to find out the inverse of the value that is on the right-hand side of the equation. The final output will be the angle between the two vectors i.e., a and b.

Vectors in a nutshell

We have described how an angle between two vectors is calculated but it does not seem an easy task. There are numerous calculations for finding the angle including the scalar product, the product of magnitude, finding magnitude by squaring, adding and again calculating the square root, which is a long and tedious process. This is the time when a "2D Vector Angle Calculator" comes into play. The calculator is very easy to use and gives the right answer within seconds. You can rely on this online calculator for finding the angle between two vectors in 2-D.