How to find dot product and cross product using vectors magnitude?

Dot product and cross product of two vectors are two different ways to calculate the product of vectors. Dot product is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.

How to find dot product and cross product using vectors magnitude?

Dot product and cross product of two vectors are two different ways to calculate the product of vectors.

Dot product is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.

On the other hand, cross product is a binary operation on two vectors in three-dimensional space.

The dot product is also known as a scalar product while cross product is also referred to as vector product.

In this post, we will discuss the easy ways to calculate dot product and cross product using the magnitude of vectors.

 To demonstrate that, we will have to know how to calculate dot product and cross product of two vectors using their formulas.

How to find dot product?

Finding dot product requires working with the dot product formula.

Dot product formula

a ∙ b = |a| × |b| × cos(θ)

In this equation,

|a| refers to the length of vector a,

|b| refers to the length of vector b,

θ is the angle between both vectors.

Now that you know the formula, let’s calculate the dot product of two vectors. Suppose there are two vectors with the following magnitude.

|a| = 5, |b| = 20, θ = 40°

Place the magnitude of both vectors in the above equation.

a · b = |a| × |b| × cos(θ)

a · b = |5| × |20| × cos(40°)

a · b  = 76.60445

In case you want to find the dot product of two vectors using the direction (coordinates), use the dot product calculator.

How to find cross product?

Calculating cross product is easy if we know the formula of the cross product.

Cross product formula

a × b = |a| |b| sin(θ) n

In this equation,

|a| refers to the length of vector a,

|b| refers to the length of vector b,

θ refers to the angle between both vectors,

n is the unit vector perpendicular to both vectors.

Let’s find the cross product of two vectors. Assume we have two vectors with the following magnitude.

|a| = 50, |b| = 40, θ = 90°

Substitute the magnitude of both vectors in the cross product formula.

a × b = |a| |b| × sin(θ) n

a × b = |50| |40| × sin(90°) n

a × b  = 2000 × 0.8939 × n

a × b  = 1788n

To find the cross product using the coordinates, you can use an online cross product calculator.

Summing up

We have calculated both types of products for the demonstration. There is not much difference in the calculations. Both deal with the magnitude of two vectors and an angle.

We may see a difference in the calculation when using coordinates, however, using the magnitudes, dot product and cross product are very simple to calculate.