# Angle between two vectors calculator

This angle between two vector calculators is really good and this tool is really easy to work with. You will learn easily how to calculate the angle between two vectors calculator.

$$Vector \ A$$
$$Vector\ B$$

#### Input

$$i=2$$ $$j=5$$ $$i=9$$ $$j=14$$

#### Solution

$$α=10.934\ Degree$$

#### Formula

$$i · j = |i| * |j| * cos(α)$$

$$Vector \ A$$
$$Vector\ B$$

#### Examples

Example For angle ϕ between the vectors

$$\mathbf{u}=\left(6, -2, 4\right) \mathbf{and\ v}=\left(-6, 4, 3\right).$$ First, calculate the dot product: $$\mathbf{u} \cdot \mathbf{v}=-32.$$ (for steps, see dot product calculator.. Next, find the lengths of the vectors: $$\left|\mathbf{u}\right|=\sqrt{\left(u_x\right)^2+\left(u_y\right)^2+\left(u_z\right)^2} =\sqrt{\left(6\right)^2+\left(-2\right)^2+\left(4\right)^2}=2 \sqrt{14}$$ $$\left|\mathbf{v}\right|=\sqrt{\left(v_x\right)^2+\left(v_y\right)^2+\left(v_z\right)^2} =\sqrt{\left(-6\right)^2+\left(4\right)^2+\left(3\right)^2}=\sqrt{61}$$ Finally, the angle is given by cos(ϕ) $$\cos\left(\phi\right)=\frac{\mathbf{u} \cdot \mathbf{v}}{\left|\mathbf{u}\right| \cdot \left|\mathbf{v}\right|}= \frac{-32}{2 \sqrt{14} \cdot \sqrt{61}}=- \frac{8 \sqrt{854}}{427}$$ $$\phi=\operatorname{acos}\left(- \frac{8 \sqrt{854}}{427}\right)= \frac{180 \operatorname{acos}{\left(- \frac{8 \sqrt{854}}{427} \right)}}{\pi}^0$$ $$\phi=\operatorname{acos}{\left(- \frac{8 \sqrt{854}}{427} \right)} rad \approx 2.15018051418339 rad$$ $$\phi=\frac{180 \operatorname{acos}{\left(- \frac{8 \sqrt{854}}{427} \right)}}{\pi}^0 \approx 123.196268653977^0$$

# About the angle between two vectors calculator tool

This angle between two vector calculators is really good and this tool is really easy to work with. You will learn easily how to calculate the angle between two vectors calculator.

With this angle between two vectors calculator, you'll quickly find out how to seek out the angle between two vectors. It doesn't matter if your vectors are in 2D or 3D, nor if their representations are coordinates or initial and terminal points - our tool may be a safe bet in every case. Play with the calculator and check the definitions and explanations below; if you're checking out the angle between two vectors formulas, you'll definitely find them there.

## The angle between two vectors formulas

Vectors represented by coordinates (standard ordered set notation, component form):

2D Vector during a component form

vectors a = [xa, ya] , b = [xb, yb]

angle = arccos[(xa * xb + ya * yb) / (√(xa2 + ya2) * √(xb2 + yb2))]

Vectors between a starting and terminal point:

2D Vector. Notation: starting and terminus ad quem coordinates.

For vector a: A = [x1, y1] , B = [x2, y2],

so vector a = [x2 - x1, y2 - y1]

For vector b: C = [x3, y3] , D = [x4, y4],

so vector b = [x4 - x3, y4 - y3]

Then insert the derived vector coordinates into the angle between two vectors formula for coordinate from point 1:

angle = arccos[((x2 - x1) * (x4 - x3) + (y2 - y1) * (y4 - y3)) / (√((x2 - x1)2 + (y2 - y1)2) * √((x4 - x3)2 + (y4 - y3)2))]

### Angle between two 3D vectors

Vectors represented by coordinates:

3D Vector during a component form

a = [xa, ya, za] , b = [xb, yb, zb]

angle = arccos[(xa * xb + ya * yb + za * zb) / (√(xa2 + ya2 + za2) * √(xb2 + yb2 + zb2))]

Vectors between a starting and terminal point: 3D Vector. Notation: starting and terminus ad quem

For vector a: A = [x1, y1, z1], B = [x2, y2, z2],

So a = [x2 - x1, y2 - y1, z2 - z1]

For vector b: C = [x3, y3, z3], D = [x4, y4, z4]

So b = [x4 - x3, y4 - y3, z4 - z3]

Find the ultimate formula analogically to the 2D version:

angle = arccos  {[(x2 - x1) * (x4 - x3) + (y2 - y1) * (y4 - y3) + (z2 - z1) * (z4 - z3)] / [√((x2 - x1)2 + (y2 - y1)2+ (z2 - z1)2) * √((x4 - x3)2 + (y4 - y3)2 + (z4 - z3)2)]}

Also, it's possible to possess one angle defined by coordinates, and therefore the other defined by a starting and terminus ad quem , but we cannot let that obscure this section even further. All that matters is that our angle between two vectors calculator has all possible combinations available to you.

How to find the angle between two vectors?

OK, the above paragraph was a touch of a TL;DR. As for how of higher understanding the formulas for the angle between two vectors, let's check where they are available from:

Start with the essential geometric formula for the dot product:

The scalar product is defined because of the product of the vectors' magnitudes multiplied by the cosine of the angle between them (here denoted by α):

a · b = |a| * |b| * cos(α)

Then, make the angle the topic of the equation:

Divide by the merchandise of the vectors' magnitudes:

cos(α) = a · b / (|a| * |b|)

Find the inverse cosine of both sides:

α = arccos[(a · b) / (|a| * |b|)]

Afterward, we'd like to brush abreast of the definition of a vectors' magnitude:

As magnitude is that the root of the sum of the vector's components to the square, we discover out that:

|v| = √(x² + y²) in 2D space

|v| = √(x² + y² + z²) in 3D space

Did you notice that it is the same formula because the one utilized in the space calculator? which it comes directly from geometry - that's, the Pythagorean theorem?

Use the algebraic formula for the scalar product (the sum of products of the vectors' components), and substitutewithin the magnitudes:

in 2D space

If vectors a = [xa, ya], b = [xb, yb], then:

α = arccos[(xa * xb + ya * yb) / (√(xa2 + ya2) * √(xb2 + yb2))]

in 3D space

If vectors a = [xa, ya, za], b = [xb, yb, zb], then:

α = arccos[(xa * xb + ya * yb + za * zb) / (√(xa2 + ya2 + za2) * √(xb2 + yb2 + zb2))]

And that's it!

Additionally, if your vectors are during a different form (you know their initial and terminal points), you will need to perform some calculations beforehand. The aim is to scale back them into standard vectors notation.

If your example vector is described by the initial point A=[x1, y1] and therefore the terminus ad quem B=[x2, y2], then vector a could also be expressed as:

a = [x2 - x1, y2 - y1]

Still not making sense? No worries! We've prepared some exemplary calculations to form sure it is a clear as crystal.

A.