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# Cross Product Calculator

Vector(A) Representation

First vector (a)

$$\vec i$$
$$\vec j$$
$$\vec k$$

Initial Point (A)

Terminal Point (B)

Vector(B) Representation

Second vector (b)

$$\vec i$$
$$\vec j$$
$$\vec k$$

Initial Point (A)

Terminal Point (B)

Table of Content
 1 What Is Cross Product? 2 Which Direction? 3 Cross Product Formula 4 Cross Product Rules of cross product of vectors 5 Cross Product 6 Dot Product 7 How to Do Cross Product of Two Vectors? 8 Cross Product Example 9 FAQ's
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An online vector cross product calculator helps you to find the cross product of two vectors and show you the step-by-step calculations. No doubt, for some individuals calculating cross product of two vectors manually looks like a daunting challenge.

Well, our calculator using the cross product formula that helps you to understand the algorithm how to do cross product of two vectors manually. This post packed with lots of information regarding cross products, so let’s begin with the basic definition of cross product.

## What Is Cross Product?

Cross product of two vectors says vector a and vector b is regarded as vector c. This is the vector that is at 90 degrees to both vectors, i.e. vector “a” as well as vector “b.” Cross product is responsible for defining the magnitude and direction of the vectors. The magnitude is not difficult to figure out as it is found to be equal to the parallelogram area. It also depicts the direction which is offered by the cross product right-hand rule.

Cross product online calculation has eased the process of cross multiplication. Now, quit worrying and just use the above vector multiplication calculator to get ease. Vector differs from scalar as scalar does not have direction while vector does have. So, if you want to find cross product of 2d, then simply try cross product vector calculator.

### Which Direction?

Remember that the cross product could point in the complete opposite direction and even still be at the right angles to the two vectors, so you have the “cross product right hand rule”

Yes, with your right-hand, you can point your index finger along vector a, and point your middle finger along vector b – you see that the cross product goes in the direction of your thumb.

### Cross Product Formula:

Vector Cross product formula is the main way for calculating the product of two vectors. The formula used for calculation of this is given as:

The cross product equation is expressed as:

C = a x b = |a| x |b| x sinθ x n

Let us discuss each element of this cross product of two vectors formula to comprehend the concept of cross multiplication more fabulously. The term “a” denotes to vectors one while the other one is depicted by “b.” The resulting vector which is the desired one to be calculated through the formula or calculator is known as vector “c.”

The angle that is formed between vector a and vector b is depicted as “θ.” Last but not least is “n” which is a unit vector – It is meant to be perpendicular to both parent vectors. Introduce components of all parent vectors to get the resultant vector. It is operated on the cross product equation mentioned above. Our cross product calculator is also uses the same formula to calculate cross product.

## About Vector Cross Product Calculator:

An online cross product calculator helps you to find the cross product of two vectors corresponding to the given coordinates or points of both vectors. In simple words, this vector product calculator allows you to find the resultant vector by multiplying two vector components and shows you the detailed step-by-step solution to your problem. Wondering to know how this online cross product of two vectors calculator works, swipe down!

### How to Calculate Cross Product Online With Cross Product Calculator:

The cross product solver is loaded with simple user-friendly interface that makes the calculation faster, and shows the cross product for the vectors within couple of seconds. Just stick to the given step, to find the cross product with this calculator:

Input:

• First of all, you have to choose a vector (A) representation, it can either by coordinates or points
• If you selected, ‘by coordinates’ option, then you ought to enter the values of coordinates
• And, if you selected, ‘by points’ option, then first, you ought to enter the initial points into the designated field of this calculator
• Very next, you have to enter the terminal points into the designated field
• For Vector (B), the above steps is same, you just have to put your required values

Output:

• Once done, the calculator shows you:
• cross product of two vectors
• Step-by-step Solution for your inputs
• Vector Magnitude
• Normalized Vector
• Spherical coordinates (Radius, Polar Angle, Azimuthal Angle)

## Cross Product Rules of cross product of vectors :

Read on and know about the simple rules and properties of cross product of vectors:

### Anti-Commutative Property:

The Cross product is an anti commutative property.

Equations:

$$\vec { A } \times \vec { B } = \vec { -B } \times \vec { A }$$
$$\vec {A} = a_1 \vec {i} + a_2 \vec {j} + a_3 \vec {k} \text{ and } \vec {B} = b_1 \vec {i} + b_2 \vec {j} + b_3 \vec {k}$$
$$\vec { A } \times \vec { B } = \begin{vmatrix} i& j& k&\\ a_1& a_2& a_3& \\ b_1& b_2& b_3& \end{vmatrix}$$
$$= (a_2b_3 – b_2a_3) \vec {i} – (a_1b_3 – b_1a_3) \vec {j} + (a_1b_2 – b_2a_1) \vec {k}$$
$$\vec { B } \times \vec { A } = \begin{vmatrix} i& j& k&\\ a_1& a_2& a_3& \\ b_1& b_2& b_3& \end{vmatrix}$$
$$= (b_2a_3 – a_2b_3) \vec {i} – (b_1a_3 – a_1b_3) \vec {j} + (b_1a_2 – a_2b_1) \vec {k}$$
$$= -[(a_2b_3 – b_2a_3) \vec {i} – (a_1b_3 – b_1a_3) \vec {j} + (a_1b_2 – b_2a_1) \vec {k}]$$
$$-\vec A \times \vec B \vec A \times \vec B = -\vec B + \vec A$$

### Distributive Property:

Yes, the cross product is having the distributive property over addition.

Equations:

$$\vec A \times (\vec B + \vec C) = \vec A \times \vec B + \vec A \times \vec C$$
$$\text{ Let } \vec {A} = a_1 \vec {i} + a_2 \vec {j} + a_3 \vec {k} , \vec {B} = b_1 \vec {i} + b_2 \vec {j} + b_3 \vec {k}$$
$$\text{ and } \vec {C} = c_1 \vec {i} + c_2 \vec {j} + c_3 \vec {k}$$
$$\vec A \times (\vec B + \vec C) = \vec A \times \vec B + \vec A \times \vec C$$
$$= \begin{vmatrix} i& j& k&\\ a_1& a_2& a_3& \\ b_1& b_2& b_3& \end{vmatrix} + \begin{vmatrix} i& j& k&\\ a_1& a_2& a_3& \\ c_1& c_2& c_3& \end{vmatrix}$$
$$= (a_2b_3 – b_2a_3) \vec {i} – (a_1b_3 – b_1a_3) \vec {j} + (a_1b_2 – a_2b_1) \vec {k}$$
$$+ (a_2c_3 – c_2a_3) \vec {i} – (a_1c_3 – c_1a_3) \vec {j} + (a_1c_2 – a_2c_1) \vec {k}$$
$$= (a_2a_3 – b_2a_3 + a_2a_3 – c_2a_3) \vec {i} – (a_1b_3 – b_1a_3 – a_1c_3 + c_1a_3) \vec {j}$$
$$+ (a_1b_2 – a_2b_1 + a_1c_2 – a_2c_1) \vec {k}$$
$$= \vec A \times (\vec B + \vec C)$$

### Jacobi Property:

Cross product is something that will satisfy the Jacobi property.

Equations:

$$\vec A \times (\vec B \times \vec C) + \vec B \times (\vec C \times \vec A) + \vec C \times (\vec A \times \vec B) = 0$$

### Zero Vector Property:

Equations:

$$a \times b = 0 \text{ if } a = 0 \text{ or } b = 0$$
$$\text{ Let } \vec {a} = 0 \vec {i} + 0 \vec {j} + 0 \vec {k} \text{ and } \vec {b} = b_1 \vec {i} + b_2 \vec {j} + b_3 \vec {k}$$
$$\text{ Then, } a \times b = \begin{vmatrix} i& j& k&\\ 0& 0& 0& \\ b_1& b_2& b_3& \end{vmatrix}$$
$$= (0 – 0) \vec{i} – (0 – 0) \vec {j} + (0 – 0) \vec {k}$$
$$= 0$$

## Cross Product vs Dot Product:

Let’s take a look!

### Cross Product:

The cross product is said to be as a vector. The magnitude of the cross product of the two vectors is referred to as the magnitude of one vector, which multiplied by the magnitude of the projection of the other vector in the direction of the orthogonal to the first vector.

If a and b are the two vectors with an angle (θ) between them, then the cross product of two vectors a and b is a x b = |a||b| sin(θ).

Yes, the direction of the resultant of the cross product of two vectors perpendicular to the plane of the two vectors, whose cross product is considering the right hand rule that we mentioned above. Our cross multiply calculator also performs the calculation by considering the same directions.

### Dot Product:

The dot product is said to be as a scalar. The dot product of two vectors provides you with the value of the magnitude of one vector multiplied by the magnitude of the projection of the other vector on the first vector.

If a and b are the two vectors with an angle (θ) between them, then the dot product of two vectors a and b is a . b = |a||b| cos(θ).

This can be considered as |a|(|b| cos(θ)), where |a| is said to be the magnitude of a and |b| cos(θ) is said to be the magnitude of the projection of b on a.

This can also be considered as |b|(|a| cos(θ)), where |b| is said to be the magnitude of b and |a| cos(θ) is said to be the magnitude of the projection of a on b.

Remember that it doesn’t make any sense to depict that ‘the dot product’s answer is limited to two dimensions’ since the outcome of the dot product of two vectors is referred to as a scalar which doesn’t have any direction. Both the dot product and cross product are part and parcel of physics.

## How to Do Cross Product of Two Vectors?

Yes, for the instant calculations of cross product you can use the cross product calculator Remember that the cross product is a type of vector multiplication that only defined in three and seven dimensions, which outputs another vector. Cross product is extremely useful for applications in physics and engineering. So, swipe down to calculate the cross product of two three-dimensional vectors defined in the Cartesian coordinates.

Calculating the Cross Product:

Step 1:

Simply, consider the two general three-dimensional vectors that are defined in Cartesian coordinates:

$$\vec a = A \vec i + B\vec j + C \vec k$$
$$\vec b = D \vec i + E\vec j + F \vec k$$

Where;

i, j, k are unit vectors, and A, B, C, D, E, F are said to be constant.

Step 2:

Now, you ought to set up the cross product matrix. The easiest ways to calculate a cross product is to set up the unit vectors with the two vectors in a matrix. Also, you can try an online matrix cross product calculator to find the cross product of the matrix.

$$\vec a \times \vec b = \begin{vmatrix} i& j& k&\\ A& B& C& \\ D& E& F& \end{vmatrix}$$

Step 3:

Now, you have to calculate the determinant of the matrix, we account cofactor expansion (expansion by minors).

$$\vec a \times \vec b = (BF – EC)\vec i – (AF – DC)\vec j + (AE – DB)\vec k$$

This vector is orthogonal to both a and b

Now let’s take a look at the cross product example!

### Cross Product Example:

Step 1:

Well, consider the two vectors below:

$$\vec u = 2\vec i – \vec j + 3\vec k$$
$$\vec v = 5\vec i + 7\vec j – 4\vec k$$

Step 2:

Now, set up the cross product matrix:

$$\vec u \times \vec v = \begin{vmatrix} i& j& k&\\ 2& -1& 3& \\ 5& 7& -4& \end{vmatrix}$$

Step 3:

Finally, calculate the determinant of the matrix:

$$\vec u \times \vec v = (4 – 21)\vec i – (-8 – 15)\vec j + (14 + 5)\vec k$$
$$= -17\vec i + 23\vec j + 19\vec k$$

## FAQ’s (Cross Product):

### Can you do cross product in 2d?

No, you can’t do at all the cross product with vectors in 2D space. Remember that the operation is not defined there. However, typically, it is interesting to determine the cross product of two vectors assuming that the 2D vectors are extended to 3D by accounting their z-coordinate to zero. Yes, this is indicating same as working with 3D vectors on the xy-plane.

### Does order matter in cross product?

In straightforward term, Yes! This is all because the cross product operation is not communicative, thus the order does matter!

### What does cross product give you?

Yes, a × b (cross product) is referred as a vector that is perpendicular (orthogonal) to both vectors (a and b), with a direction that is given by the right-hand rule and a magnitude is equal to the area of the parallelogram that the vectors span.

### What is triple cross product?

The product of three vectors is said to be as the triple product. In simple words, the cross product of one vector with the cross product of another two vectors said to be as triple cross product.

If you have three vectors A, B, and C, then the vector triple product is indicated as:

A × (B × C) = (A . C) B − (A . B) C

(A × B) × C = −C × (A × B) = −(C . B) A + (C . A) B

### Why is cross product not commutative?

The cross product doesn’t follow the commutative property as the direction of the unit vector becomes opposite when the vector product occurs in a reverse manner. However, both the cross products of both the vectors (a and b) in both the possible ways – that is said to be as AxB and BxA are additive inverse of each other.

### What is cross product used for?

The cross product is used to determine a vector that is perpendicular to the plane spanned by two vectors. On the other hand, the dot product is used to determine the length of a vector or the angle between two vectors.

### Why is J negative in cross product?

According to Laplace’s expansion for the determinant, when it comes to the geometrical point of view, the cross product is corresponds to the signed area of the parallelogram that is with the two vectors as sides, you can readily determine the minus (-) sign in its expressions by symbolic determinant that indeed needs a minus (-) sign for the →j coordinate.

### How do you know if a cross product is orthogonal?

No doubt the cross product is always orthogonal to both vectors and with magnitude (0) zero when these vectors are referred to as parallel and has maximum magnitude ‖a‖‖b‖ when these vectors are referred to as the orthogonal.

### How do you cross multiply?

To cross multiply, you have to stick to the given steps:

• First of all, you have to start by multiplying the numerator of the left-hand fraction by the denominator of the right-hand fraction
• Very next, you ought to multiply the numerator of the right-hand fraction by the denominator of the left-hand fraction
• Now, you ought to set the 2 products equal to each other
• Finally, you have to solve for the variable

### Is cross product sin or cos?

In calculating terms, the dot product of two unit vectors are yields the cosine (that may be positive or negative) of the angle between these two unit vectors. And, when it comes to the cross product, the magnitude of the two unit vectors yields the sine (that is always said to be positive).

## Wrapping it Up!

Remember that the concept of the cross product is taken into account to describe the product of physical quantities that have both a magnitude and a direction associated with them. So, use this online vector cross product calculator to solve the 2d vectors problem.

## References:

From Wikipedia, the free encyclopedia – cross product definition – Computing the cross product – properties of cross product – Alternative ways to compute the cross product – Cross product as an external product – cross product matrix

Recently updated from the source of wikihow – How to Calculate the Cross Product of Two Vectors – cross product example

From the source of mathsisfun (advanced) – Cross product formula – the ways of calculating cross product – (cross product right hand rule) – Dot product

From the recent source of study – lesson 12 in chapter 2 of the course – Finding The Cross Product of Two Vectors – Equation for Cross Product