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128 LearnersLast updated on September 2, 2025
Cross Product Calculator
Calculators are reliable tools for solving simple mathematical problems and advanced calculations like trigonometry. Whether you're cooking, tracking BMI, or planning a construction project, calculators will make your life easy. In this topic, we are going to talk about cross product calculators.
What is a Cross Product Calculator?
A cross product calculator is a tool to compute the cross product of two vectors in three-dimensional space. The cross product is a vector that is perpendicular to both of the vectors being multiplied and has many applications in physics and engineering. This calculator makes the calculation much easier and faster, saving time and effort.
How to Use the Cross Product Calculator?
Given below is a step-by-step process on how to use the calculator:
Step 1: Enter the components of the first vector: Input the x, y, and z components of the first vector into the given fields.
Step 2: Enter the components of the second vector: Input the x, y, and z components of the second vector.
Step 3: Click on calculate: Click the calculate button to find the cross product of the two vectors.
Step 4: View the result: The calculator will display the resulting vector instantly.
How to Calculate the Cross Product?
In order to calculate the cross product of two vectors, the calculator uses the following formula:
If vector A = (a1, a2, a3) and vector B = (b1, b2, b3), then the cross product A × B = (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1).
The resulting vector is perpendicular to both A and B.
Tips and Tricks for Using the Cross Product Calculator
When using a cross product calculator, there are a few tips and tricks to make it easier and avoid mistakes:
Remember that the cross product is only defined in three-dimensional space.
Ensure that you input the vector components correctly to avoid errors.
Understand that the magnitude of the cross product represents the area of the parallelogram formed by the original vectors.
Common Mistakes and How to Avoid Them When Using the Cross Product Calculator
We may think that when using a calculator, mistakes will not happen. But it is possible for users to make mistakes when using a calculator.
Mistake 1
Incorrectly inputting vector components.
Double-check the vector components before calculating. Input errors can lead to incorrect results.
Mistake 2
Not recognizing zero vectors.
If the cross product is a zero vector, it means the original vectors are parallel or one is a zero vector.
Mistake 3
Misinterpreting the result direction.
The resulting vector's direction is perpendicular to the plane formed by the original vectors. Ensure you understand its orientation.
Mistake 4
Ignoring the order of vectors.
The cross product is not commutative. A × B is not the same as B × A; the result will have the opposite direction.
Mistake 5
Assuming the cross product can be used in 2D space.
The cross product is specifically for 3D vectors. For 2D vectors, use the dot product instead.
Cross Product Calculator Examples
Problem 1
What is the cross product of vectors A = (2, 3, 4) and B = (5, 6, 7)?
Use the formula:
A × B = (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1)
A × B = (3*7 - 4*6, 4*5 - 2*7, 2*6 - 3*5)
A × B = (21 - 24, 20 - 14, 12 - 15)
A × B = (-3, 6, -3)
Explanation
By calculating each component, we find the cross product to be (-3, 6, -3).
Problem 2
Find the cross product of vectors C = (1, 0, 0) and D = (0, 1, 0).
Use the formula:
C × D = (0*0 - 0*1, 0*0 - 0*0, 1*1 - 0*0)
C × D = (0, 0, 1)
Explanation
The resulting vector (0, 0, 1) is along the z-axis, perpendicular to both C and D.
Problem 3
Calculate the cross product of vectors E = (2, -1, 3) and F = (4, 0, -2).
Use the formula:
E × F = (-1*(-2) - 3*0, 3*4 - 2*(-2), 2*0 - (-1)*4)
E × F = (2, 12 + 4, 0 + 4)
E × F = (2, 16, 4)
Explanation
The cross product of E and F is (2, 16, 4), showing the perpendicular vector.
Problem 4
Determine the cross product for vectors G = (0, 2, 1) and H = (1, 0, 3).
Use the formula:
G × H = (2*3 - 1*0, 1*1 - 0*3, 0*0 - 2*1)
G × H = (6, 1, -2)
Explanation
The cross product is (6, 1, -2), perpendicular to both G and H.
Problem 5
What is the cross product of vectors I = (3, -2, 5) and J = (1, 4, -1)?
Use the formula:
I × J = (-2*(-1) - 5*4, 5*1 - 3*(-1), 3*4 - (-2)*1)
I × J = (2 - 20, 5 + 3, 12 + 2)
I × J = (-18, 8, 14)
Explanation
The resulting vector from the cross product of I and J is (-18, 8, 14).
FAQs on Using the Cross Product Calculator
1.How do you calculate the cross product?
2.Can the cross product be zero?
3.What does the cross product represent?
4.Is the cross product commutative?
5.What are the applications of the cross product?
Glossary of Terms for the Cross Product Calculator
- Cross Product: A vector operation on two vectors in three-dimensional space resulting in a third vector that is perpendicular to the plane of the input vectors.
- Vector: A quantity having direction as well as magnitude, especially as determining the position of one point in space relative to another.
- Perpendicular: Two lines or vectors at an angle of 90 degrees to each other.
- Magnitude: The length or size of a vector.
- Parallelogram: A four-sided figure with opposite sides parallel, used in the context of the cross product to describe the area formed by two vectors.
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Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
