Cross Product Calculator

What is a Cross Product

The cross product, sometimes referred to as the vector product in mathematics, is a binary operation of two vectors in three dimensions. The “Cross Product” results from the two vectors, a and b. It can only be expressed in three dimensions and is symbolized by “a b.” The product of the two vectors, denoted by the letter “c,” is perpendicular to both vectors a and b.

Cross Product Formula

The formula to calculate the cross products:

\( a × b = |a| |b| sin (θ) n \)

Where,

• \( | a | \) and \( | b | \) denote the vector lengths
• θ is the vectors’ angle (0° to 180°).
• n denotes the unit vector perpendicular to vectors a and b.
• \( | \space | \) displays the magnitude

Cross Product of Two Vectors

Cross product is a type of vector multiplication carried out between two vectors of various forms or natures. For example, the cross product and dot product can be used to multiply two or more vectors. The resultant vector is the cross-product of two vectors or the vector product.

Properties of Cross product

The cross product has four properties, and the first two are the most crucial. These are:

1- Anti-commutative Property

This property primarily exhibits negative signs.

\( \overrightarrow{A} × \overrightarrow{B} = -\overrightarrow{B} × \overrightarrow{A} \)

The anti-commutativity takes part in the coordinate notation method, so the equalities i, j and k become -i, -j, and -k, respectively.

2- Distributive Property

It demonstrates how to resolve equations like a \( (b + c) \).

\( \overrightarrow{A} × (\overrightarrow{B} + \overrightarrow{C}) = \overrightarrow{A} × \overrightarrow{B} + \overrightarrow{A} × \overrightarrow{C} \)

3- Jacobi Property

Jacobi property, named after the German Mathematician Carl Gustav Jakob, is represented.

\( \overrightarrow{A} × (\overrightarrow{B} × \overrightarrow{C}) + \overrightarrow{B} × (\overrightarrow{C} × \overrightarrow{A}) + \overrightarrow{C} × (\overrightarrow{A} × \overrightarrow{B}) = 0 \)

4- Zero Vector Property

Represented as:

\( a × b = 0 \) if \( a = 0 \) or \( b = 0 \)

There are three distinct sub-properties to it.

1. An identity that is additive, i.e., a + 0 Equals a.
2. Zero multiplied by any spatial vector has this feature; hence 0 (a) = 0.
3. Every vector in space is at right angles (orthogonal), which means that a. 0 = 0.

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