Changes

{{math-h}}

{{math-h}}

The '''cross product''' (or vector product) of two [[vector|vectors]] in 3-space is itself a vector in 3-space and is written <math>a\times b</math>. The magnitude of the resulting vector is

The '''cross product''' (or vector product) of two [[vector|vectors]] in 3-space is itself a vector in 3-space and is written <math>\vec{a}\times \vec{b}</math>. The magnitude of the resulting vector is

<math>

<math>

|a\times b|=|a| |b| \sin\theta

|\vec{a}\times \vec{b}|=|\vec{a}| |\vec{b}| \sin\theta

</math>

</math>

where <math>\theta</math> is the angle between the two vectors.  

where <math>\theta</math> is the angle between the two vectors.  

For two vectors <math>a=(a_1,a_2,a_3)</math> and <math>b=(b_1,b_2,b_3)</math>,

For two vectors <math>\vec{a}=(a_1,a_2,a_3)</math> and <math>\vec{b}=(b_1,b_2,b_3)</math>,

<math>a\times b = (a_2b_3-a_3b_2, a_3b_1-a_1b_3, a_1b_2-a_2b_1)</math>.

<math>\vec{a}\times \vec{b} = (a_2b_3-a_3b_2, a_3b_1-a_1b_3, a_1b_2-a_2b_1)</math>.

The direction of the cross product is normal to both of the vectors '''a''' and '''b'''. Since there are two such directions the chosen one is defined by the [[right hand rule]]: with your right hand, point your fingers along the direction of the first vector and curl them towards the second vector. The direction your thumb points gives the direction of the cross product. The cross product is a convenient way to find the volume of a [[parallelepiped]]. One may simply take the cross product of two legs and then find the dot product of that vector with the remaining leg (assuming that the legs are vectors). So, a [[parallelepiped]] with sides <math>a,b</math> and <math>c</math> will have a volume,<math>V</math>, of:

The direction of the cross product is normal to both of the vectors <math>\vec{a}</math> and <math>\vec{b}</math>. Since there are two such directions the chosen one is defined by the [[right hand rule]]: with your right hand, point your fingers along the direction of the first vector and curl them towards the second vector. The direction your thumb points gives the direction of the cross product. The cross product is a convenient way to find the volume of a [[parallelepiped]]. One may simply take the cross product of two legs and then find the dot product of that vector with the remaining leg (assuming that the legs are vectors). So, a [[parallelepiped]] with sides <math>\vec{a},\vec{b}</math> and <math>\vec{c}</math> will have a volume,<math>V</math>, of:

:<math>V = |a \cdot (b\times c)|.</math>

:<math>V = |\vec{a} \cdot (\vec{b}\times \vec{c})|.</math>

== Application ==

== Application ==