Cross Product Calculator

Cross Product Calculator

Select the parameters for both the vectors and write their unit vector coefficients to determine the cross product, normalized vector, and spherical coordinates, with detailed calculations shown

Vector(A) Representation:

First vector (a)

$$\vec i$$

$$\vec j$$

$$\vec k$$

Vector(B) Representation:

Second vector (b)

$$\vec i$$

$$\vec j$$

$$\vec k$$

The 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.

What Is Cross Product?

The cross product of two vectors involves vectors a and b resulting in a new vector c. This vector c is perpendicular (at 90°) to both a and b. The cross product determines both the magnitude and the direction of the resulting vector. The magnitude corresponds to the area of the parallelogram formed by a and b, while the direction follows the right-hand rule.

Cross Product Formula:

The cross product formula is used to calculate the product of two vectors. It is expressed as:

C = a × b = |a| × |b| × sinθ × n

How to Calculate Cross Product Using Our Calculator:

The cross product calculator features a user-friendly interface to quickly compute the cross product of two vectors. Follow these steps to use it:

Input:

Select the representation of Vector A, either by coordinates or by points.
If using coordinates, enter the x, y, z values of the vector.
If using points, enter the initial point in the designated field.
Enter the terminal point in the designated field.
Repeat the above steps for Vector B, entering the required values.

Output:

The calculator provides:
Cross product of the two vectors
Step-by-step solution for your inputs
Vector magnitude
Normalized vector
Spherical coordinates (radius, polar angle, azimuthal angle)

How to Do Cross Product of Two Vectors?

Calculating the Cross Product:

Step 1:

Consider two general three-dimensional vectors defined in Cartesian coordinates:

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

Here, i, j, k are unit vectors, and A, B, C, D, E, F are constants.

Step 2:

Set up the cross product matrix. The easiest way to calculate a cross product is using a matrix with the unit vectors and the components of the two vectors. You can also use an online matrix cross product calculator to simplify the calculation.

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

Step 3:

Calculate the determinant of the matrix using cofactor expansion (expansion by minors):

$$ \vec a \times \vec b = (BF - CE)\vec i - (AF - CD)\vec j + (AE - BD)\vec k $$

The resulting vector is orthogonal (perpendicular) to both a and b. Now, let’s look at an example.

Cross Product Example:

Step 1: Consider the two vectors:

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

Step 2: 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: Calculate the determinant of the matrix:

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

References:

From Wikipedia – cross product definition and properties. Recently updated from WikiHow – How to Calculate the Cross Product of Two Vectors – example of cross product calculation.