Vector Cross Product Calculator

In math, you can perform calculations to get a cross product of 2 arbitrary vectors. To make it easier for you, this cross product calculator can do the calculations for you. Rather than perform the calculations yourself which can be a very tedious process, this calculator only requires some values for it to generate the final results. Read on to learn more about this vector cross product calculator and some common questions on the same subject.

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How to use the cross product calculator? 

Just like most online calculators, this one is simple and easy to use. Instead of manual computations, this vector multiplication calculator will provide you with the cross products in a matter of seconds. Here are the steps for using it:

  • First, input the 3 values for vector a (x, y, z).
  • Then input the 3 values for vector b (x, y, z).
  • After entering all of the values, you will automatically get the cross products of vector c (x, y, z)


What is the cross product method? 

Both manual computation and the cross product calculator use a different process than the cross product method. This method involves a comparison of two fractions. In this method, you multiply the numerator of the first fraction with the denominator of the second one and vice versa. Then you compare your answers to see whether the fractions are equal or not.

The cross product method is a type of shortcut which allows you to find the common denominator without changing the value of the fractions. Of course, using this method means that you can only compare two fractions at a time. If you need to compare more, you have to keep repeating the process.


What is a cross product of two vectors? 

The cross product of two vectors which are non-collinear and arbitrary, vector a and vector b is a vector c which has a perpendicular relationship with both vectors. Therefore, if you have a plane where vector a and vector b rest on, vector c will be normal to the said plane.

You can determine the direction of vector c using your hand, and the right-hand rule. If the middle and index fingers of your right-hand point towards the direction of vector a and vector b, your thumb will show the direction of vector c. Aside from using this vector cross product calculator, you can manually compute with the following formula: 

c = a × b = |a| * |b| * sinθ * n,


a and b refer to the arbitrary vectors

|a| and |b| refer to the magnitudes of the arbitrary vectors

c refers to the resulting cross product vector

θ refers to the angle between the vectors

n refers to a unit vector that’s perpendicular to the plane and determined by vector a and vector b

When performing manual computations, you can also use this formula:

a × b = (a₂b₃ – a₃b₂) * i + (a₃b₁ – a₁b₃) * j + (a₁b₂ – a₂b₁) * k


How to determine direction of cross product? 

A dot product stands for the similarity of a vector with a single number. Calculate for the dot product with the following formula: 

dot product = (ax, ay, az) * (bx, by, bz) = ax bx + ay by + az bz = ||vec{a}|| ||vec{b}|| cos(ϴ)

To calculate a candidate for the cross product, use this formula:

a candidate for the cross product = amount of difference = ||vec{a}|| ||vec{b}|| sin(ϴ)

But looking at these formulas, you’ll realize that there are some missing details. Therefore, you need to express these differences as a vector while keeping these points in mind:

  • The cross product’s size refers to the numerical value of the “amount of difference” while sin(θ) is the percentage.
  • The basis of the cross product’s direction is both of the inputs which means that it doesn’t favor either of them.
  • The resulting vector separately represents x * y and x * z, although these both differ from x.
  • You can determine a plane using 2 vectors and the resulting cross product would point in a direction that’s different from those vectors.

Now to determine the direction of the resulting vector, you can use the right-hand rule. To do this, hold your thumb and your first two fingers out. Your two fingers point to the direction of the vectors while your thumb points in the direction of the resulting cross point.


What is cross product in physics? 

Before we discuss what a cross product is in physics, let’s have a rundown of everything we’ve discussed so far. The cross product also known as the vector in physics of vector a and vector be can be symbolically written as: 

𝑎 × 𝑏 or 𝑎 ∧ 𝑏.

By definition, the cross product is a vector with a magnitude:

| 𝑎 × 𝑏 | = | 𝑎 | |𝑏 | sin𝜃


𝜃 refers to the angle between vector 𝑎 and vector 𝑏

The direction of 𝑎 × 𝑏 is a vector that’s perpendicular to both vector a and vector b. The best way to determine that direction is by using the right-hand rule. You can also see this concept in physics such as in how a screw moves. If you twist a screw from, then it would start moving forward.

Alternatively, if you hold out your thumb and your first two fingers and position them at right angles from each other, there would be a change in the formula. This time, your thumb and first finger would point in the same direction while your second finger would point to a different direction or the direction of 𝑎 × 𝑏. Cross products or vector products frequently occur in physics.

You can even use physics to help you determine the direction of the cross product as we’ve illustrated in our two examples:

  • If you twist a screw from 𝑎 to 𝑏, it would move upward
  • If you position the first three fingers of your hand at right angles, the direction of 𝑎 to 𝑏 would be that of your second finger