Dot Product Calculator
Enter values to find the dot product of two vectors with dot product calculator.
Dot product calculator calculates the dot product of two vectors a and b in Euclidean space. Enter i, j, and k for both vectors to get scalar number.
a . b
Vector dot product calculator shows step by step scalar multiplication.
What is dot product?
Dot product is an algebraic operation that takes two equal-length sequences of numbers usually coordinate vectors, and returns a single number.
Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.
a . b usually read as a dot b.
Dot product formula
Use this equation to calculate dot product of two vectors if magnitude (length) is given.
a ∙ b = |a| × |b| × cos(θ)
Where
|a| is length of vector a
|b| is length of vector b
θ is the angle between a and b
Vector Directions
We can also find dot product by using the direction of both vectors.
(ai aj ak) ∙ (bi bj bk) = (ai ∙ bi + aj ∙ bj + ak ∙ bk)
Where
i, j, and k refers to x, y, and z coordinates on Cartesian plane.
How to find dot product of two vectors?
The dot product of two vectors can be calculated by using the dot product formula.
Method 1 – Vector Direction
Vector a = (2i, 6j, 4k)
Vector b = (5i, 3j, 7k)
Place the values in the formula.
a ∙ b = (2, 6, 4) ∙ (5, 3, 7)
(ai aj ak) ∙ (bi bj bk) = (ai ∙ bi + aj ∙ bj + ak ∙ bk)
(2 6 4) ∙ (5 3 7) = (2 ∙ 5 + 6 ∙ 3 + 4 ∙ 7)
(2 6 4) ∙ (5 3 7) = (10 + 18 + 28)
a ∙ b = 56
Method 2 – Vector Magnitude
|a| = 15, |b| = 10, θ = 30°
Place the values in the formula.
a · b = |a| × |b| × cos(θ)
a · b = 15 × 10 × cos(30°)
a · b = 23.14
References:
Dot Product Formula from tutorial.math.lamar.edu.