Difference between revisions of "The Cross Product"

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[[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>.
 +
The cross product of <math> \mathbf{u} </math> and <math> \mathbf{v} </math> can also be written in determinant form like so:
  
<math>\begin{vmatrix}
+
<math>\mathbf{w} = det\begin{vmatrix}
 
i & j & k\\
 
i & j & k\\
 
u_1 & u_2 & u_3\\
 
u_1 & u_2 & u_3\\

Revision as of 17:25, 20 September 2021

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 and , the cross product of and (notated as ) is . The cross product of and can also be written in determinant form like so:

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