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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.

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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)

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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?

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 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

How to Calculate Cross Product With Our 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 cross product 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)

How to Do Cross Product of Two Vectors?

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:

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

References:

From Wikipedia, the free encyclopedia – cross product definition – Computing the cross product – properties of cross product.

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