Late product

Spat, which is spanned by three vectors

The late product , also called mixed product , is the scalar product of the cross product of two vectors and a third vector. It gives the oriented volume of by the three vectors spanned spats (parallelepiped). Its amount is therefore equal to the volume of the opened spatula. The sign is positive if these three vectors form a legal system in the given order ; if they form a left system, it is negative. If the three vectors lie in one plane, their late product is zero.

In Cartesian coordinates, the late product can also be calculated using the determinant formed from the three vectors .

definition

The triple product of three vectors , and the three-dimensional Euclidean vector space can be defined as follows: ${\displaystyle ({\vec {a}},{\vec {b}},{\vec {c}})}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {c}}}$ ${\displaystyle \mathbb {R} ^{3}}$

${\displaystyle ({\vec {a}},{\vec {b}},{\vec {c}})=({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}}$.

notation

Often no separate notation is introduced for the late product, but you just write . Other common values are: , and . ${\displaystyle ({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}}$ ${\displaystyle [{\vec {a}},{\vec {b}},{\vec {c}}]}$ ${\displaystyle \langle {\vec {a}},{\vec {b}},{\vec {c}}\rangle }$ ${\displaystyle |{\vec {a}}\ {\vec {b}}\ {\vec {c}}|}$

properties

• The late product is not commutative . However, the value does not change if you swap the factors cyclically :

${\displaystyle ({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}=({\vec {b}}\times {\vec {c}})\cdot {\vec {a}}=({\vec {c}}\times {\vec {a}})\cdot {\vec {b}}}$.

• The late product can be calculated using the determinant. For

${\displaystyle {\vec {a}}={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}},\ {\vec {b}}={\begin{pmatrix}b_{1}\\b_{2}\\b_{3}\end{pmatrix}},\ {\vec {c}}={\begin{pmatrix}c_{1}\\c_{2}\\c_{3}\end{pmatrix}}}$

applies

${\displaystyle ({\vec {a}},{\vec {b}},{\vec {c}})=\det {\begin{pmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{pmatrix}}=\det {\begin{pmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{pmatrix}}}$.

The proof can be provided, for example, by simple calculation, see below.

• Since the vectors in the late product can be exchanged cyclically and the scalar product is commutative, the following applies

${\displaystyle ({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}={\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})}$.

So you can "swap" the two arithmetic symbols with appropriately adapted brackets (which would otherwise be nonsensical).

• In contrast to the cyclical exchange, a sign change occurs when two factors are exchanged:

${\displaystyle ({\vec {a}},{\vec {b}},{\vec {c}})=-({\vec {b}},{\vec {a}},{\vec {c}})}$.

• Further applies because of : ${\displaystyle {\vec {a}}\times {\vec {a}}={\vec {0}}}$

${\displaystyle ({\vec {a}},{\vec {a}},{\vec {b}})=0}$.

• Multiplication by a scalar is associative : ${\displaystyle \alpha \in \mathbb {R} }$

${\displaystyle (\alpha \cdot {\vec {a}},{\vec {b}},{\vec {c}})=\alpha \cdot ({\vec {a}},{\vec {b}},{\vec {c}})}$.

• A distributive law applies :

${\displaystyle ({\vec {a}},{\vec {b}},{\vec {c}}+{\vec {d}})=({\vec {a}},{\vec {b}},{\vec {c}})+({\vec {a}},{\vec {b}},{\vec {d}})}$.

Geometric interpretation

Amount of volume and oriented volume

The volume of the space spanned by the three vectors (parallelepiped) is equal to the amount of the space product: ${\displaystyle V}$ ${\displaystyle {\vec {a}},{\vec {b}},{\vec {c}}}$

${\displaystyle V=|({\vec {a}},{\vec {b}},{\vec {c}})|=|({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}|}$.

If you do not form the amount, you get the oriented volume. The (irregular) tetrahedron spanned by the 3 vectors has the volume of the spatula. ${\displaystyle {\tfrac {1}{6}}}$

Derivation

The volume of a spat is calculated from the product of its base area and its height .

${\displaystyle V=A_{g}\cdot h}$

The cross product is the normal vector on the base area spanned through and , which forms a right-handed coordinate system with and and the amount of which is equal to the area of ​​the parallelogram spanned through and , that is . ${\displaystyle {\vec {a}}\times {\vec {b}}}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle A_{g}=|{\vec {a}}\times {\vec {b}}|}$

The height of the spat is the projection of the vector onto the direction of this normal vector (its unit vector ). If these include the angle , the definition of the scalar product applies ${\displaystyle {\vec {c}}}$ ${\displaystyle \alpha }$

${\displaystyle h=|{\vec {c}}|\cos \alpha ={\hat {e}}_{{\vec {a}}\times {\vec {b}}}\cdot {\vec {c}}}$.

It follows

${\displaystyle V=A_{g}\cdot h=|{\vec {a}}\times {\vec {b}}|({\hat {e}}_{{\vec {a}}\times {\vec {b}}}\cdot {\vec {c}})=({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}}$.

The volume is zero for 90 °, i.e. if the vectors lie in one plane. They are then called coplanar and linearly dependent . ${\displaystyle \alpha }$

The oriented volume is negative if it is greater than 90 °. Then the vector product and the projected height point in opposite directions because the vectors form a left system. ${\displaystyle \alpha }$

Derivation of the algebraic properties

The late product can also be derived with the Levi-Civita symbol . To do this, the scalar product is first represented by a sum:

${\displaystyle ({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}=\sum _{i=1}^{3}({\vec {a}}\times {\vec {b}})_{i}\cdot c_{i}.}$

The cross product is now represented with the Levi-Civita symbol using a sums notation:

${\displaystyle \sum _{i=1}^{3}({\vec {a}}\times {\vec {b}})_{i}\cdot c_{i}=\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}a_{j}b_{k}c_{i}.}$

The totally antisymmetric epsilon tensor is equal or equal . The late product can thus be expressed as follows: ${\displaystyle \varepsilon _{ijk}}$ ${\displaystyle \varepsilon _{kij}}$ ${\displaystyle \varepsilon _{jki}}$

${\displaystyle \sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}a_{j}b_{k}c_{i}=\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{kij}a_{j}b_{k}c_{i}=\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{jki}a_{j}b_{k}c_{i}.}$

The summation symbols can be swapped. In addition, you can now use brackets:

${\displaystyle \sum _{i=1}^{3}\left(\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}a_{j}b_{k}\right)c_{i}=\sum _{k=1}^{3}\left(\sum _{i=1}^{3}\sum _{j=1}^{3}\varepsilon _{kij}c_{i}a_{j}\right)b_{k}=\sum _{j=1}^{3}\left(\sum _{k=1}^{3}\sum _{i=1}^{3}\varepsilon _{jki}b_{k}c_{i}\right)a_{j}.}$

If you write the cross products again without the Levi-Civita symbol , you get the desired identity:

${\displaystyle ({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}=({\vec {c}}\times {\vec {a}})\cdot {\vec {b}}=({\vec {b}}\times {\vec {c}})\cdot {\vec {a}}.}$

Word origin

The term Spatprodukt goes back to the term "Spat" for a parallelepiped (parallelepiped, parallelotope). In geology, the suffix -spat indicates that the mineral in question can be easily cleaved. Examples: feldspar , calcite . These spades have clear break lines. In particular, the crystals of calcite are very similar to the geometric ideal of a parallelepiped. Calculating the volume of such a parallelepiped or spade results in the term spar product.

literature

• Wolfgang Gawronski: Basics of Linear Algebra. Aula-Verlag, Wiesbaden 1996, ISBN 3-89104-566-2 .

Web links

• Late product calculator : Calculates the late product of three vectors.