The vector dot product calculator comes in handy when you are solving vector multiplication problems. Instead of calculating the dot product by hand, you can simply input the components of two vectors into this tool and let it do the math for you. Keep reading to learn what dot product formula our calculator uses, or switch to the cross product calculator.
Vector multiplication types
There are two main types of vector multiplication: the dot product (also called the scalar product), denoted with the symbol "·", and the cross product, denoted with the symbol "×". The main difference is that the product of the dot operation is a single number, while the product of the cross operation is a vector.
What is the dot product formula?
Let's assume that all our calculations will be performed in a 3D space. That means that every vector can be written down using three components:
a = [a₁, a₂, a₃]
b = [b₁, b₂, b₃]
Geometrically, the dot product is described as the product of the vectors' magnitudes multiplied by the cosine of the angle between them. It can be expressed as an equation:
a·b = |a| * |b| * cosθ
If you are not sure what is the magnitude of a vector or how to calculate it, head to the unit vector calculator for more details on the subject.
You can probably notice that if the angle between two vectors is equal to 90°, then the scalar product will always be equal to 0, regardless of the vectors' magnitudes. Similarly, if the angle is equal to 0° (the vectors are collinear), then the dot product is found by multiplying the multitudes only.
Algebraically, it is a sum of products of the vectors' components. For three-component vectors, the dot product formula looks as follows:
a·b = a₁ * b₁ + a₂ * b₂ + a₃ * b₃
In a space that has more than three dimensions, you simply need to add more terms to the summation. If, on the other hand, you want to multiply vectors in a 2D space, you have to omit the third term of the formula.
Determining the vector dot product
So, how does our vector multiplication calculator work? Follow this step-by-step example to get a better understanding of the principle behind this process.
- Choose your vector a. For example, we will take a = [4, 5, -3].
- Choose your vector b. Let's assume it is equal to b = [1, -2, -2].
- Calculate the product of the first components of each vector. In this case, it is equal to
4 * 1 = 4.
- Calculate the product of the second (middle) components of each vector. In this case, it is equal to
5 * (-2) = -10.
- Calculate the product of the third components of each vector. In this case, it is equal to
(-3) * (-2) = 6.
- Add all of these results together to find the dot product of the vectors a and b.
4 + (-10) + 6 = 0
The result is 0. This is the scalar product of these two vectors. It means that they are perpendicular to each other (the angle between them is equal to 90°).