Cross Products

The cross product is an essential concept in vector mathematics, especially in physics and engineering. It involves the multiplication of two vectors to produce a new vector that is perpendicular to both input vectors. The cross product is typically used to calculate the area of a parallelogram, the torque applied to an object, and the orientation of magnetic and electric fields.

How to calculate the cross product

To calculate the cross product of two vectors A and B in three-dimensional space, we use the following formula:

A x B = (A_y * B_z - A_z * B_y) i - (A_x * B_z - A_z * B_x) j + (A_x * B_y - A_y * B_x) k

where A_x, A_y, and A_z are the components of vector A, and B_x, B_y, and B_z are the components of vector B. The resulting cross product is a new vector with components i, j, and k.

Cross product properties

The cross product is anticommutative, meaning:

A x B = - (B x A)

The cross product is distributive:

A x (B + C) = A x B + A x C

The scalar multiplication property:

(kA) x B = A x (kB) = k(A x B)

where k is a scalar.

Cross product example

Given two vectors

A = (2, 3, 4)

and

B = (5, 6, 7)

, we can calculate their cross product as follows:

A x B = (3 * 7 - 4 * 6) i - (2 * 7 - 4 * 5) j + (2 * 6 - 3 * 5) k
= (-3) i - (-6) j + (-3) k
=

-3, 6, -3>

In this example, the cross product of vectors A and B is the vector

-3, 6, -3>

Topics related to the Cross Products

Vectors

Solving Systems of Linear Equations

Systems of Linear Equations

Flashcards covering the Cross Products

Calculus 3 Flashcards

Calculus AB Flashcards

Practice tests covering the Cross Products

Calculus 3 Diagnostic Tests

Multivariable Calculus Practice Tests

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