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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

by Points(A) by Coordinates:

First vector (a)

$$\vec i$$

$$\vec j$$

$$\vec k$$

by Points(B) by Coordinates:

Second vector (b)

$$\vec i$$

$$\vec j$$

$$\vec k$$

Related

The cross product calculator allows you to compute the cross product of two vectors while providing detailed step-by-step calculations. Manually calculating the cross product can be challenging, but this tool simplifies the process.

What Is Cross Product?

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

Cross Product Formula:

The cross product of vectors is calculated using the formula:

C = a × b = |a| × |b| × sinθ × n

How to Calculate Cross Product Using Our Calculator:

The cross product calculator has a simple interface to quickly compute the cross product. Follow these steps:

Input:

Choose how to represent Vector A: by coordinates or by points.
If using coordinates, enter the x, y, z values.
If using points, enter the initial and terminal points in the designated fields.
Repeat the process for Vector B.

Output:

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

How to Compute the Cross Product of Two Vectors

Step-by-Step Calculation:

Step 1:

Consider two three-dimensional vectors 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:

Construct the cross product matrix. Using a determinant simplifies 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 using cofactor expansion:

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

The resulting vector is perpendicular to both a and b. Next, let's see a practical example.

Cross Product Example:

Step 1: Consider the 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: Compute the determinant:

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

References:

From Wikipedia – definition and properties of cross product. Updated from WikiHow – step-by-step example of cross product calculation.

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