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Angle Between Two Vectors Calculator

Angle Between Two Vectors Calculator

Vector(A) Representation

First vector (a)

$$\vec x$$
$$\vec y$$
$$\vec z$$

Initial Point (A)

Terminal Point (B)

Vector(B) Representation

Second vector (b)

$$\vec x$$
$$\vec y$$
$$\vec z$$

Initial Point (A)

Terminal Point (B)

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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 shortest angle at which one of the vectors rotates to a position consistent with the other vector. Remember that vector quantities have both magnitude and direction. Vectors can be expressed in two-dimensional and three-dimensional spaces.

Angle Between Two Vectors Formula:

There are different formulas that are used by the angle between two vectors calculator which depend on vector data:

Find Angle between Two 2d Vectors:

  • Vectors represented by coordinates

Vectors \(m = [x_m, y_m] , n = [x_n, y_n]\)

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

  • The vector between terminal points

For vector p: M = \([x_m, y_m] , \text { N} = [x_n, y_n]\),

so vector p = \([x_n – x_m, y_n – y_m]\)

For vector q: C = \([x_c, y_c] , \text { D} = [x_d, y_d]\),

so vector q = \([x_d – x_c, y_d – y_c]\)

Then angle between vectors calculator substitute the vector coordinates into the angle between two vectors formula for the point A:

$$= cos^{-1}[\frac{((x_n – x_m) * (x_d – x_c) + (y_n – y_m) * (y_d – y_c))}{(\sqrt{((x_n – x_m)^2 + (y_n – y_m)^2)} * \sqrt{((x_d – x_c)^2 + (y_d – y_c)^2)}}]$$

However, an online Arccos Calculator allows you to calculate the inverse of the cosine of a certain number.

Find Angle between Two 3d Vectors:

The vector angle calculator use the following aspects for finding the angle between two vectors.

  • Vectors represented by coordinates:

$$m = [x_m, y_m, z_m] , n = [x_n, y_n, z_n]$$

$$angle = cos^{-1}[\frac{(xm * xn + ym * yn + zm * zn)}{(\sqrt{(xm^2 + ym^2 + zm^2)} * \sqrt{(x_n^2 + y_n^2 + z_n^2)}}]$$

  • Vectors between a terminal point:

For vector a: M = \([x_m, y_m, z_m], \text { N} = [x_n, y_n, z_n]\),

so a = \([x_n – x_m, y_n – y_m, z_n – z_m]\)

For vector b: O = \([x_o, y_o, z_o], P = [x_p, y_p, z_p]\)

so b = \([x_p – x_o, y_p – y_o, z_p – z_o]\)

Find the formula analogically to the 2D version:

$$angle = cos^{-1}{[(xn – xm) * (xp – xo) + (yn – ym) * (yp – yo) + (zn – zm) * (zp – zo)] / [\sqrt{((xm – xn)^2 + (yn – ym)^2+ (zn – zm)^2)} * \sqrt{((xp – xo)^2 + (yp – yo)^2 + (zp – zo)^2)}]}$$

You can determine one angle by coordinates, and the other by a terminal point with the angle between two vectors calculator.

How to Find the Angle between Two Vectors:

You can find the angle between vectors with the understanding of different formulas, let’s check where they come from:

  • The formula for the dot product:

The dot product is expressed as the product of the vectors’ sizes or magnitudes multiplied by the cos of the angle between them (here denoted by θ):

$$a · b = |a| x |b| cos(θ)$$

  • Now, the angle of the equation:

Divide by the product of two vectors’ magnitudes:

$$cos(θ) = a · b / (|a| x |b|)$$

Then, find the cos inverse of both sides:

$$θ = cos^{-1}[(a · b) / (|a| x |b|)]$$

  • As magnitude is the square root (\(\sqrt{ }\)) of the sum of the components to the second power:

Vector in 2D space:

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

Vector in 3D space

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

  • Then, the angle between two vectors calculator uses the formula for the dot product, and substitute it in the magnitudes:

2D space

If vectors m = \([x_m, y_m], n = [x_n, y_n]\), then:

$$θ = cos^{-1}[(x_m * x_n + y_m * y_n) / (√(x_m^2 + y_m^2) * √(x_n^2 + y_n^2))]$$

3D space

If vectors m = \([x_m, y_m, z_m], n = [x_n, y_n, z_n]\), then:

$$θ = cos^{-1}[(x_m * x_n + y_m * y_n + z_m * z_n) / (\sqrt{(x_m^2 + y_m^2 + z_m^2)} * \sqrt{(x_n^2 + y_n^2 + z_n^2)})]$$

These formulas are used by angle between vectors calculator for two and three dimensional vectors magnitude.

However, use an online free Cosine Calculator that helps you in calculating the cosine value of the given angle in degrees and radians.

Example: find angle between two 3d vectors

A = {4, 6, 8}

B = {3, 2, 5}

Now start with dot product of A and B, A.B

$$A ⋅B =Ax . Bx + Ay . By + Az . Bz$$

$$A ⋅B =(4 . 3) + (6 . 2) + (8 . 5)$$

$$A ⋅B = (12) + (12) + (40)$$

$$A ⋅B =64$$

Magnitude of Vector A:

$$|A|=\sqrt{Ax^2+Ay^2+Az^2}$$

$$|A|=\sqrt{(4)2+(6)2+(8)2}$$

$$|A|=\sqrt{16+36+64}$$

$$|A|=10.77033$$

Magnitude of Vector B:

$$|B|=\sqrt{Bx2+By2+Bz2}$$

$$|B|=\sqrt{(3)2+(2)2+(5)2}$$

$$|B|=\sqrt{9+4+25}$$

$$|B|=6.16441$$

Angle between vectors A and B:

$$cosθ=(A. B)/(|A||B|)$$

$$cosθ=(64)/(10.77033∗6.16441)$$

$$cosθ=(64)/(66.39277)$$

$$cosθ=0.9639604$$

$$θ=15.42911 deg$$

However, you can get exact values by plugs the same values in the angle between vectors calculator.

How angle between two vectors calculator Works?

The angle between vector calculator find the angle θ separating two Vectors A and B in two and three-dimensional space with these steps:

Input:

  • First, select the 2D or 3D dimension of vectors.
  • Now, choose the vector representation (by Coordinates or Terminal points) from the drop-down list.
  • Then, plugs all given values in the related fields.
  • To see the results, click on the “Calculate” button.

Output:

  • The angle between two vectors calculator provides stepwise calculations for the Dot product, magnitude, and angle between vectors.
  • You can find the 2D and 3D vectors numerous times as per requirements by clicking on recalculate button.

FAQ:

Define the angle between two antiparallel vectors?

Antiparallel vectors are parallel vectors in opposite directions. The angle between these two vectors is 180°.

What is the magnitude of the unit vector?

Since the unit vector is 1 by definition, if you want to use the unit vector in the A direction, you must divide by this magnitude.

Is the Angle a vector quantity?

In fact, the angle is a dimensionless vector. It has direction and magnitude. We can measure clockwise and counterclockwise angles based on their rotation behavior. Therefore, this angle is indeed a “vector quantity”.

Conclusion:

Use this angle between two vectors calculator to determine the angle between vector components. The concept of vector angle is used to describe the angular difference of the physical quantity assigned to the quantity and direction. The vector angle is calculated from the first endpoint. This free online calculator provides dot product and magnitude quickly with 100% accuracy.

Reference:

From the source of Wikipedia: Algebraic definition, Geometric definition, Scalar projection, and first properties, Equivalence of the definitions, Application to the law of cosines, Vector algebra relations, Magnitudes.

From the source of Wiki How: Finding the Angle Between Two Vectors, Defining the Angle Formula, write the Law of Cosines for this triangle.

From the source of Krista King Math: Angle Between Two Vectors, Formula for the angle between vector, How to calculate the angle, Finding the angle between two vectors in three dimensions.