Difference between revisions of "The Cross Product"
Jump to navigation
Jump to search
| Line 1: | Line 1: | ||
[[File:Cross product parallelogram.svg|thumb|Cross_product_parallelogram]] | [[File:Cross product parallelogram.svg|thumb|Cross_product_parallelogram]] | ||
The cross product is an operation between two 3-dimensional vectors that returns a third vector orthogonal (i.e., perpendicular) to the first two. For vectors <math> \mathbf{u} = \langle u_1, u_2, u_3 \rangle </math> and <math> \mathbf{v} = \langle v_1, v_2, v_3 \rangle </math>, the cross product of <math> \mathbf{u} </math> and <math> \mathbf{v} </math> (notated as <math> u \times v </math>) is <math> \mathbf{w} = \langle u_2v_3 - u_3v_2, -(u_1v_3 - u_3v_1), u_1v_2 - u_2v_1 \rangle</math>. | The cross product is an operation between two 3-dimensional vectors that returns a third vector orthogonal (i.e., perpendicular) to the first two. For vectors <math> \mathbf{u} = \langle u_1, u_2, u_3 \rangle </math> and <math> \mathbf{v} = \langle v_1, v_2, v_3 \rangle </math>, the cross product of <math> \mathbf{u} </math> and <math> \mathbf{v} </math> (notated as <math> u \times v </math>) is <math> \mathbf{w} = \langle u_2v_3 - u_3v_2, -(u_1v_3 - u_3v_1), u_1v_2 - u_2v_1 \rangle</math>. | ||
| − | + | One way to remember the cross product of <math> \mathbf{u} </math> and <math> \mathbf{v} </math> is to calculate it with the following determinant: | |
<math>\mathbf{w} = \textrm{det}\begin{vmatrix} | <math>\mathbf{w} = \textrm{det}\begin{vmatrix} | ||
| Line 7: | Line 7: | ||
u_1 & u_2 & u_3\\ | u_1 & u_2 & u_3\\ | ||
v_1 & v_2 & v_3\\ | v_1 & v_2 & v_3\\ | ||
| − | \end{vmatrix} </math> | + | \end{vmatrix} = |
| + | (u_2v_3 - u_3v_2) - (u_1v_3 - u_3v_1) + (u_1v_2 - u_2v_1) | ||
| + | </math> | ||
==Resources== | ==Resources== | ||
* [https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product The Cross Product], OpenStax | * [https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product The Cross Product], OpenStax | ||
* [https://omega0.xyz/omega8008/calc3/cross-product-dir/cornell-lecture.html Cross Product], Cornell University | * [https://omega0.xyz/omega8008/calc3/cross-product-dir/cornell-lecture.html Cross Product], Cornell University | ||
Revision as of 17:29, 20 September 2021
The cross product is an operation between two 3-dimensional vectors that returns a third vector orthogonal (i.e., perpendicular) to the first two. For vectors and , the cross product of and (notated as ) is . One way to remember the cross product of and is to calculate it with the following determinant:
Resources
- The Cross Product, OpenStax
- Cross Product, Cornell University
