Four dimensional cross product of THREE vectors
$begingroup$
There are many MSE posts about how to define a cross product in $mathbb{R^4}$. It is impossible to define a cross product of two vectors in $mathbb{R^4}$, since there are infinitely many directions perpendicular to those two vectors, and we don't know which direction to choose. However, If we are given THREE vectors $A,B,C$, it is possible to find a unique direction perpendicular to this three vectors, if $A,B,C$ are independent. However, finding this perpendicular vector involves solving a system of equations.
So my question is:can we define a Quasi Cross Product ${A,B,C}$ on $mathbb{R^4}$, so that we can find a direction perpendicular to $A,B,C$ without solving a system of equations?
linear-algebra systems-of-equations cross-product
$endgroup$
add a comment |
$begingroup$
There are many MSE posts about how to define a cross product in $mathbb{R^4}$. It is impossible to define a cross product of two vectors in $mathbb{R^4}$, since there are infinitely many directions perpendicular to those two vectors, and we don't know which direction to choose. However, If we are given THREE vectors $A,B,C$, it is possible to find a unique direction perpendicular to this three vectors, if $A,B,C$ are independent. However, finding this perpendicular vector involves solving a system of equations.
So my question is:can we define a Quasi Cross Product ${A,B,C}$ on $mathbb{R^4}$, so that we can find a direction perpendicular to $A,B,C$ without solving a system of equations?
linear-algebra systems-of-equations cross-product
$endgroup$
add a comment |
$begingroup$
There are many MSE posts about how to define a cross product in $mathbb{R^4}$. It is impossible to define a cross product of two vectors in $mathbb{R^4}$, since there are infinitely many directions perpendicular to those two vectors, and we don't know which direction to choose. However, If we are given THREE vectors $A,B,C$, it is possible to find a unique direction perpendicular to this three vectors, if $A,B,C$ are independent. However, finding this perpendicular vector involves solving a system of equations.
So my question is:can we define a Quasi Cross Product ${A,B,C}$ on $mathbb{R^4}$, so that we can find a direction perpendicular to $A,B,C$ without solving a system of equations?
linear-algebra systems-of-equations cross-product
$endgroup$
There are many MSE posts about how to define a cross product in $mathbb{R^4}$. It is impossible to define a cross product of two vectors in $mathbb{R^4}$, since there are infinitely many directions perpendicular to those two vectors, and we don't know which direction to choose. However, If we are given THREE vectors $A,B,C$, it is possible to find a unique direction perpendicular to this three vectors, if $A,B,C$ are independent. However, finding this perpendicular vector involves solving a system of equations.
So my question is:can we define a Quasi Cross Product ${A,B,C}$ on $mathbb{R^4}$, so that we can find a direction perpendicular to $A,B,C$ without solving a system of equations?
linear-algebra systems-of-equations cross-product
linear-algebra systems-of-equations cross-product
asked Dec 2 '18 at 8:25
Ma JoadMa Joad
824316
824316
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3 Answers
3
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oldest
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$begingroup$
You have the "same" determinant formula. If $vec{a} = (a_1,a_2,a_3,a_4)$, similarly for $vec{b}$ and $vec{c}$, then $$vec{a}timesvec{b}times vec{c} = begin{vmatrix} vec{e}_1 & vec{e}_2 & vec{e}_3 & vec{e}_4 \ a_1 & a_2 & a_3 & a_4 \ b_1 & b_2 & b_3 & b_4 \ c_1 & c_2 & c_3 & c_4end{vmatrix},$$where $(vec{e}_1,ldots,vec{e}_4)$ is the standard basis for $Bbb R^4$. This does not require solving a system. Example: $$(1,1,0,0)times (0,1,1,0) times (0,0,1,1) = begin{vmatrix} vec{e}_1 & vec{e}_2 & vec{e}_3 & vec{e}_4 \ 1 & 1 & 0 & 0 \ 0 & 1 & 1 & 0 \ 0 & 0 & 1 & 1end{vmatrix} = (1,-1,1,-1).$$
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1
$begingroup$
An interested person could check this for determinant $vec x = vec a times vec b times vec c$ by computing the three dot products $vec x circ vec a$, $vec x circ vec b$, $vec x circ vec c$ and ensuring they are all zero.
$endgroup$
– DanielV
Dec 2 '18 at 8:36
add a comment |
$begingroup$
Yes, it is the Hodge dual of the wegde product. Given three vectors $A=(a_1,a_2,a_3,a_4)$, $B=(b_1,b_2,b_3,b_4)$, and $C=(c_1,c_2,c_3,c_4)$, the triple-product is defined as
$${A,B,C}= (det M_1, det M_2, det M_3, det M_4)$$
with
$$ M_1 = begin{pmatrix} a_2 & b_2 & c_2 \
a_3 & b_3 & c_3 \
a_4 & b_4 & c_4 end{pmatrix} $$
where the other matrices $M_{2}$, $M_3$, $M_4$ are obtained by cyclically permuting the indices. Note that the product is antisymmetric with respect to exchanging any of its arguments.
$endgroup$
add a comment |
$begingroup$
The short answer is yes. One way is to take the formal determinant
$$left|begin{matrix}e_1&e_2&e_3&e_4\
a_1&a_2&a_3&a_4\
b_1&b_2&b_3&b_4\
c_1&c_2&c_3&c_4\
end{matrix}right|$$
where $e_1,ldots,e_4$ are the standard unit vectors, and $a=sum a_ie_i$
etc., are the three vectors.
Or you can rephrase this in terms
of exterior powers and the Hodge star operator.
All this works in $n$ dimensions too.
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You have the "same" determinant formula. If $vec{a} = (a_1,a_2,a_3,a_4)$, similarly for $vec{b}$ and $vec{c}$, then $$vec{a}timesvec{b}times vec{c} = begin{vmatrix} vec{e}_1 & vec{e}_2 & vec{e}_3 & vec{e}_4 \ a_1 & a_2 & a_3 & a_4 \ b_1 & b_2 & b_3 & b_4 \ c_1 & c_2 & c_3 & c_4end{vmatrix},$$where $(vec{e}_1,ldots,vec{e}_4)$ is the standard basis for $Bbb R^4$. This does not require solving a system. Example: $$(1,1,0,0)times (0,1,1,0) times (0,0,1,1) = begin{vmatrix} vec{e}_1 & vec{e}_2 & vec{e}_3 & vec{e}_4 \ 1 & 1 & 0 & 0 \ 0 & 1 & 1 & 0 \ 0 & 0 & 1 & 1end{vmatrix} = (1,-1,1,-1).$$
$endgroup$
1
$begingroup$
An interested person could check this for determinant $vec x = vec a times vec b times vec c$ by computing the three dot products $vec x circ vec a$, $vec x circ vec b$, $vec x circ vec c$ and ensuring they are all zero.
$endgroup$
– DanielV
Dec 2 '18 at 8:36
add a comment |
$begingroup$
You have the "same" determinant formula. If $vec{a} = (a_1,a_2,a_3,a_4)$, similarly for $vec{b}$ and $vec{c}$, then $$vec{a}timesvec{b}times vec{c} = begin{vmatrix} vec{e}_1 & vec{e}_2 & vec{e}_3 & vec{e}_4 \ a_1 & a_2 & a_3 & a_4 \ b_1 & b_2 & b_3 & b_4 \ c_1 & c_2 & c_3 & c_4end{vmatrix},$$where $(vec{e}_1,ldots,vec{e}_4)$ is the standard basis for $Bbb R^4$. This does not require solving a system. Example: $$(1,1,0,0)times (0,1,1,0) times (0,0,1,1) = begin{vmatrix} vec{e}_1 & vec{e}_2 & vec{e}_3 & vec{e}_4 \ 1 & 1 & 0 & 0 \ 0 & 1 & 1 & 0 \ 0 & 0 & 1 & 1end{vmatrix} = (1,-1,1,-1).$$
$endgroup$
1
$begingroup$
An interested person could check this for determinant $vec x = vec a times vec b times vec c$ by computing the three dot products $vec x circ vec a$, $vec x circ vec b$, $vec x circ vec c$ and ensuring they are all zero.
$endgroup$
– DanielV
Dec 2 '18 at 8:36
add a comment |
$begingroup$
You have the "same" determinant formula. If $vec{a} = (a_1,a_2,a_3,a_4)$, similarly for $vec{b}$ and $vec{c}$, then $$vec{a}timesvec{b}times vec{c} = begin{vmatrix} vec{e}_1 & vec{e}_2 & vec{e}_3 & vec{e}_4 \ a_1 & a_2 & a_3 & a_4 \ b_1 & b_2 & b_3 & b_4 \ c_1 & c_2 & c_3 & c_4end{vmatrix},$$where $(vec{e}_1,ldots,vec{e}_4)$ is the standard basis for $Bbb R^4$. This does not require solving a system. Example: $$(1,1,0,0)times (0,1,1,0) times (0,0,1,1) = begin{vmatrix} vec{e}_1 & vec{e}_2 & vec{e}_3 & vec{e}_4 \ 1 & 1 & 0 & 0 \ 0 & 1 & 1 & 0 \ 0 & 0 & 1 & 1end{vmatrix} = (1,-1,1,-1).$$
$endgroup$
You have the "same" determinant formula. If $vec{a} = (a_1,a_2,a_3,a_4)$, similarly for $vec{b}$ and $vec{c}$, then $$vec{a}timesvec{b}times vec{c} = begin{vmatrix} vec{e}_1 & vec{e}_2 & vec{e}_3 & vec{e}_4 \ a_1 & a_2 & a_3 & a_4 \ b_1 & b_2 & b_3 & b_4 \ c_1 & c_2 & c_3 & c_4end{vmatrix},$$where $(vec{e}_1,ldots,vec{e}_4)$ is the standard basis for $Bbb R^4$. This does not require solving a system. Example: $$(1,1,0,0)times (0,1,1,0) times (0,0,1,1) = begin{vmatrix} vec{e}_1 & vec{e}_2 & vec{e}_3 & vec{e}_4 \ 1 & 1 & 0 & 0 \ 0 & 1 & 1 & 0 \ 0 & 0 & 1 & 1end{vmatrix} = (1,-1,1,-1).$$
answered Dec 2 '18 at 8:31
Ivo TerekIvo Terek
45.7k952141
45.7k952141
1
$begingroup$
An interested person could check this for determinant $vec x = vec a times vec b times vec c$ by computing the three dot products $vec x circ vec a$, $vec x circ vec b$, $vec x circ vec c$ and ensuring they are all zero.
$endgroup$
– DanielV
Dec 2 '18 at 8:36
add a comment |
1
$begingroup$
An interested person could check this for determinant $vec x = vec a times vec b times vec c$ by computing the three dot products $vec x circ vec a$, $vec x circ vec b$, $vec x circ vec c$ and ensuring they are all zero.
$endgroup$
– DanielV
Dec 2 '18 at 8:36
1
1
$begingroup$
An interested person could check this for determinant $vec x = vec a times vec b times vec c$ by computing the three dot products $vec x circ vec a$, $vec x circ vec b$, $vec x circ vec c$ and ensuring they are all zero.
$endgroup$
– DanielV
Dec 2 '18 at 8:36
$begingroup$
An interested person could check this for determinant $vec x = vec a times vec b times vec c$ by computing the three dot products $vec x circ vec a$, $vec x circ vec b$, $vec x circ vec c$ and ensuring they are all zero.
$endgroup$
– DanielV
Dec 2 '18 at 8:36
add a comment |
$begingroup$
Yes, it is the Hodge dual of the wegde product. Given three vectors $A=(a_1,a_2,a_3,a_4)$, $B=(b_1,b_2,b_3,b_4)$, and $C=(c_1,c_2,c_3,c_4)$, the triple-product is defined as
$${A,B,C}= (det M_1, det M_2, det M_3, det M_4)$$
with
$$ M_1 = begin{pmatrix} a_2 & b_2 & c_2 \
a_3 & b_3 & c_3 \
a_4 & b_4 & c_4 end{pmatrix} $$
where the other matrices $M_{2}$, $M_3$, $M_4$ are obtained by cyclically permuting the indices. Note that the product is antisymmetric with respect to exchanging any of its arguments.
$endgroup$
add a comment |
$begingroup$
Yes, it is the Hodge dual of the wegde product. Given three vectors $A=(a_1,a_2,a_3,a_4)$, $B=(b_1,b_2,b_3,b_4)$, and $C=(c_1,c_2,c_3,c_4)$, the triple-product is defined as
$${A,B,C}= (det M_1, det M_2, det M_3, det M_4)$$
with
$$ M_1 = begin{pmatrix} a_2 & b_2 & c_2 \
a_3 & b_3 & c_3 \
a_4 & b_4 & c_4 end{pmatrix} $$
where the other matrices $M_{2}$, $M_3$, $M_4$ are obtained by cyclically permuting the indices. Note that the product is antisymmetric with respect to exchanging any of its arguments.
$endgroup$
add a comment |
$begingroup$
Yes, it is the Hodge dual of the wegde product. Given three vectors $A=(a_1,a_2,a_3,a_4)$, $B=(b_1,b_2,b_3,b_4)$, and $C=(c_1,c_2,c_3,c_4)$, the triple-product is defined as
$${A,B,C}= (det M_1, det M_2, det M_3, det M_4)$$
with
$$ M_1 = begin{pmatrix} a_2 & b_2 & c_2 \
a_3 & b_3 & c_3 \
a_4 & b_4 & c_4 end{pmatrix} $$
where the other matrices $M_{2}$, $M_3$, $M_4$ are obtained by cyclically permuting the indices. Note that the product is antisymmetric with respect to exchanging any of its arguments.
$endgroup$
Yes, it is the Hodge dual of the wegde product. Given three vectors $A=(a_1,a_2,a_3,a_4)$, $B=(b_1,b_2,b_3,b_4)$, and $C=(c_1,c_2,c_3,c_4)$, the triple-product is defined as
$${A,B,C}= (det M_1, det M_2, det M_3, det M_4)$$
with
$$ M_1 = begin{pmatrix} a_2 & b_2 & c_2 \
a_3 & b_3 & c_3 \
a_4 & b_4 & c_4 end{pmatrix} $$
where the other matrices $M_{2}$, $M_3$, $M_4$ are obtained by cyclically permuting the indices. Note that the product is antisymmetric with respect to exchanging any of its arguments.
answered Dec 2 '18 at 8:31
FabianFabian
19.7k3674
19.7k3674
add a comment |
add a comment |
$begingroup$
The short answer is yes. One way is to take the formal determinant
$$left|begin{matrix}e_1&e_2&e_3&e_4\
a_1&a_2&a_3&a_4\
b_1&b_2&b_3&b_4\
c_1&c_2&c_3&c_4\
end{matrix}right|$$
where $e_1,ldots,e_4$ are the standard unit vectors, and $a=sum a_ie_i$
etc., are the three vectors.
Or you can rephrase this in terms
of exterior powers and the Hodge star operator.
All this works in $n$ dimensions too.
$endgroup$
add a comment |
$begingroup$
The short answer is yes. One way is to take the formal determinant
$$left|begin{matrix}e_1&e_2&e_3&e_4\
a_1&a_2&a_3&a_4\
b_1&b_2&b_3&b_4\
c_1&c_2&c_3&c_4\
end{matrix}right|$$
where $e_1,ldots,e_4$ are the standard unit vectors, and $a=sum a_ie_i$
etc., are the three vectors.
Or you can rephrase this in terms
of exterior powers and the Hodge star operator.
All this works in $n$ dimensions too.
$endgroup$
add a comment |
$begingroup$
The short answer is yes. One way is to take the formal determinant
$$left|begin{matrix}e_1&e_2&e_3&e_4\
a_1&a_2&a_3&a_4\
b_1&b_2&b_3&b_4\
c_1&c_2&c_3&c_4\
end{matrix}right|$$
where $e_1,ldots,e_4$ are the standard unit vectors, and $a=sum a_ie_i$
etc., are the three vectors.
Or you can rephrase this in terms
of exterior powers and the Hodge star operator.
All this works in $n$ dimensions too.
$endgroup$
The short answer is yes. One way is to take the formal determinant
$$left|begin{matrix}e_1&e_2&e_3&e_4\
a_1&a_2&a_3&a_4\
b_1&b_2&b_3&b_4\
c_1&c_2&c_3&c_4\
end{matrix}right|$$
where $e_1,ldots,e_4$ are the standard unit vectors, and $a=sum a_ie_i$
etc., are the three vectors.
Or you can rephrase this in terms
of exterior powers and the Hodge star operator.
All this works in $n$ dimensions too.
answered Dec 2 '18 at 8:32
Lord Shark the UnknownLord Shark the Unknown
103k1160132
103k1160132
add a comment |
add a comment |
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