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Cross Product Calculator

Use our free Best Cross Product Calculator to check the product of two vectors instantly. All you need to do is enter your first vector and second vector in the provided input boxes and press the calculate button to get the product of given two vectors as result in the output box within no time.

Cross Product Calculator
a = i + j + k
b = i + j + k

Cross Product Calculator: Are you searching for any handy tool that does all calculations and provides the cross product of vectors simply? Then, here is the best way for you. Utilize our Cross Product Calculator and get the answer for your vectors easily with lengthy calcuations at a faster pace. Have a look at the below sections and explore more about Cross Product of Vectors, its formula, an example using cross product and step by step procedure to solve the vectors.

Step by Step Procedure to Solve Vector Cross Product

Cross Product is a method used to multiply any two vectors. Here, we are giving the simple steps to calculate the cross product of vectors. Follow these steps and solve your vectors effortlessly.

  • Take any two vectors of same order.
  • Represent the given two vectors as matrix. Where first row is a unit vector, second row is our first vector and the third row is the second vector. .
  • To check the vector cross product, we need to perform the determinant of matrix.
  • Expand the matrix and find the det matrix value.
  • The formed coefficients of the unit vector will be the cross product of two vectors.

Cross Product of Vectors Formula

The formula to compute the cross product of two different vectors is given below:

a x b = │a││b│ sin(θ) n


a, b are the two different vectors

θ is the angle between those vectors

n is the unit matrix.


Question: Calculate the cross product between a=(3,−3,1) and b=(4,9,2)?


Given vectors are

a=(3,−3,1) and b=(4,9,2)

The matrix will be

             i j k

a x b = 3 -3 1

             4 9 2

│a x b │ = i(-3*2 - 1*9) -j(3*2-1*4)+ k(3*9 + 3*4)

= i(-6-9) -j(6-4)+ k(27+12)

= -15i-2j+39k

a x b = (-15, -2, 39)

(3,−3,1) x (4,9,2) = (-15, -2, 39)

If you want to simplify cross product of vectors without performing any mathematical operations, them have a look at and hit the Cross Product Calculator link under the vector section and get the accurate answer without fail.

Cross Product Calculator

FAQs on Cross Product Calculator

1. What do you mean by Cross Product?

Cross product is a binary operation of two vectors in a three dimensional space. It is also called as vector product. The formula to find the cross product of vectors is a x b = │a││b│ sin(θ) n

Where a,b are the two vectors

θ is the angle between two vectors

││ is the magnitude

n is the unit vector.

2. What happens when you cross product of identical vector?

When you calculate the cross product of identical vectors, it produces a degenerate parallelogram with no area. The cross product will become zero.

3. Calculate the area of the parallelogram spanned by the vectors a=(3,−3,1) and c=(−12,12,−4).

Vectors are

a=(3,−3,1) and c=(−12,12,−4)

│a x c│ = i j k

              3 -3 1

              -12 12 -4

=i(-3x-4 -12x1) + j(-12x1 - 3x-4) +k(3x12 - (-3)x(-12)

=i(12-12)+ j(-12+12)+ k(36-36)


The magnitude of zero vector is zero, so the area of the parallelogram is zero.

4. What is cross product used for?

The cross product is used to find a vector which is perpendicular to the plane spanned by two different vectors. It has many applications in physics when dealing with the rotating bodies.

5. Find the angle between two vector a and b, where a =(-4, 3, 0) and b =(2, 0, 0)?

The formula to get the angle between two vectors is

sin(θ) = a x b / │a││b│

θ = sin-1(a x b / │a││b│)

First calculate a x b

a x b = i(0) -j(0)+k(-6)


│a│ = √-42+32+02

=√16+9 = √25 = 5

│b│ = √22+02+02


We know that θ = sin-1(a x b / │a││b│)

θ = sin-1(-6 / 5x2)

=sin-1 (3/5) = 36.87°

Hence, the angle is 36.87°.