Vector multiplication: How to apply a formula of Cross product and dot product?

Vectors multiplication can be performed in several ways. Prominently, there are two ways you can multiply vectors.

  1. Cross product
  2. Dot Product

Even though the cross product and dot product can be easily calculated using a cross product calculator and dot product calculator respectively, we will explain the method to manually calculate both of them.

In this post, you will learn vector multiplication formulas, definitions of cross and dot products, and how to calculator cross product as well as dot product by using those formulas.

What is Cross Product?

Wikipedia defines cross product as,

The cross product is a binary operation on two vectors in three-dimensional space{\displaystyle \mathbb {R} ^{3}} and is denoted by the symbol.{\displaystyle \times } Given two linearly independent vectors a and b, the cross product a × b is a vector that is perpendicular to both a and b.

Cross product

It has many applications in computer programming, physics, mathematics, and engineering. Cross product is also referred to as vector product.

Cross Product Formula

The formula used for calculating cross product is,

a × b = |a| |b| sin(θ) n


|a| is the length of vector a,

|b| is the length of vector b,

θ is the angle between a and b,

n is the unit vector at right angles to both a and b.

Cross product formula with points

You can also calculate the cross product of vectors using this equation.

(ai aj ak) × (bi bj bk) = (aj × bk – ak × bj    ak × bi – ai × bk    ai  bj – aj × bi)


 i, j, and k represent x, y, and z coordinates on the Cartesian plane.

What is Dot Product?

According to BetterExplained,

The dot product also referred to as scalar product is an algebraic operation that takes two equal-length sequences of numbers, and returns a single number.

In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is extensively used.

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Dot Product Formula

The formula to find dot product is,

a . b = |a| |b| cos (θ)


Cross product

|a| is the length of vector a,

|b| is the length of vector b,

θ is the angle between a and b.

Dot product formula with points

We can also find dot product by using the points of both vectors.

(aaj ak) . (bi bj bk) = (ai . bi + aj . bj + ak . bk)


 i, j, and k refer to x, y, and z coordinates on the Cartesian plane.