Calculator Use

3-Dimensional Vectors Angle Calculator is a free online tool that displays the Angle of 3-Dimensional Vectors. This online 3-Dimensional Vectors Angle Calculator tool performs the calculation faster, and it displays the result in a fraction of seconds.

The procedure to use the 3-Dimensional Vectors Angle Calculator is as follows:

Step 1: Enter a values in the input field

Step 2: Now click the "Calculate" button to get the result

Step 3: Finally, The Angle of 3-Dimensional Vectors will be displayed in the output field

How to calculate the angle between two vectors?

Vector: a=(x₁,y₁,z₁), b=(x₂,y₂,z₂)，Then，a.b=(x₁x₂+y₁y₂+z₁z₂)

|a|=√(x₁²+y₁²+z₁²)，|b|=√(x₂²+y₂²+z₂²)

cos α =(x₁x₂+y₁y₂+z₁z₂)/[√(x₁²+y₁²+z₁²)*√(x₂²+y₂²+z₂²)]

Solved Example:

Vector coordinates a: < 4, 3, 2 >

Vector coordinates b: < 1, 2, 5 >

cos α = a*b/[|a|*|b|]

Step by step solution:

1) Calculate the modulus (length) of the first and second vectors:

|a| = √[x₁²+y₁²+z₁²] =√( 4² + 5²+²) = √(16 + 9+4) = √29

|b| = √[x₂²+y₂²+z₂²] = √(1² + 4²+5²)= √(1 + 4 + 25) = √30

2) Calculate the product of the modules of vectors:

|a| ⋅ |b| =√(29 ⋅30) =√870

3) Calculate the scalar product of vectors: a and b

a*b = (x₁,y₁,z₁)(x₂,y₂,z₂) = x₁x₂+y₁y₂+z₁z₂ = 4 ⋅ 1 + 3 ⋅2 + 2 ⋅ 5 = 20

) Calculate the cosine of the angle between the vectors:

cos α = a*b/[|a|*|b|] = 20 / √870 = 0.677

5) Calculate the value of the angle ∠α between the vectors:

∠α = 32.60º