Recent changes to Vectors

Vectors modified by Vladimir Chukharev

Vladimir Chukharev — Mon, 12 Nov 2018 07:57:09 -0000

--- v45
+++ v46
@@ -38,8 +38,8 @@
 (10)  *\[\[a×b\]×c\] = b ·(a⋅c) - a ·(b⋅c)*

 Other useful equations:
-  (11) *\[a×b\]⋅\[c×d\] = (a⋅c)⋅(b⋅d)-(a⋅d)⋅(b⋅c)*
-  (12) *\[a×b\]×\[c×d\] = (a⋅\[b×d\])⋅c-(a⋅\[b×c\])⋅d = (a⋅\[c×d\])⋅b-(b⋅\[c×d\])⋅a*
+  (11) *\[a×b\]⋅\[c×d\] = (a⋅c)⋅(b⋅d) - (a⋅d)⋅(b⋅c)*
+  (12) *\[a×b\]×\[c×d\] = (a⋅\[b×d\])⋅c - (a⋅\[b×c\])⋅d = (a⋅\[c×d\])⋅b - (b⋅\[c×d\])⋅a*
   (13) *\[a×b\]×\[a×c\] = (a⋅\[b×c\])⋅a*

 Components over a general linear coordinate basis a, b, c  can be determined according to the equation

Vectors modified by Vladimir Chukharev

Vladimir Chukharev — Mon, 12 Nov 2018 07:54:17 -0000

--- v44
+++ v45
@@ -39,7 +39,7 @@

 Other useful equations:
   (11) *\[a×b\]⋅\[c×d\] = (a⋅c)⋅(b⋅d)-(a⋅d)⋅(b⋅c)*
-  (12) *\[a×b\]×\[c×d\] = (a⋅\[b×d\])⋅c-(a⋅\[b×c\])⋅d=(a⋅\[c×d\])⋅b-(b⋅\[c×d\])⋅a*
+  (12) *\[a×b\]×\[c×d\] = (a⋅\[b×d\])⋅c-(a⋅\[b×c\])⋅d = (a⋅\[c×d\])⋅b-(b⋅\[c×d\])⋅a*
   (13) *\[a×b\]×\[a×c\] = (a⋅\[b×c\])⋅a*

 Components over a general linear coordinate basis a, b, c  can be determined according to the equation

Vectors modified by Vladimir Chukharev

Vladimir Chukharev — Mon, 12 Nov 2018 07:47:05 -0000

--- v43
+++ v44
@@ -38,14 +38,14 @@
 (10)  *\[\[a×b\]×c\] = b ·(a⋅c) - a ·(b⋅c)*

 Other useful equations:
-  (11) *\[a×b\]⋅\[c×d\]=(a⋅c)⋅(b⋅d)-(a⋅d)⋅(b⋅c)*
-  (12) *\[a×b\]×\[c×d\]=(a⋅\[b×d\])⋅c-(a⋅\[b×c\])⋅d=(a⋅\[c×d\])⋅b-(b⋅\[c×d\])⋅a*
-  (13) *\[a×b\]×\[a×c\]=(a⋅\[b×c\])⋅a*
+  (11) *\[a×b\]⋅\[c×d\] = (a⋅c)⋅(b⋅d)-(a⋅d)⋅(b⋅c)*
+  (12) *\[a×b\]×\[c×d\] = (a⋅\[b×d\])⋅c-(a⋅\[b×c\])⋅d=(a⋅\[c×d\])⋅b-(b⋅\[c×d\])⋅a*
+  (13) *\[a×b\]×\[a×c\] = (a⋅\[b×c\])⋅a*

 Components over a general linear coordinate basis a, b, c  can be determined according to the equation
-  (14) *d=((d⋅\[b×c\])⋅a+(a⋅\[d×c\])⋅b+(a⋅\[b×d\])⋅c)/(a⋅\[b×c\])*
+  (14) *d = ((d⋅\[b×c\])⋅a + (a⋅\[d×c\])⋅b + (a⋅\[b×d\])⋅c)/(a⋅\[b×c\])*
 where *a⋅\[b×c\]!=0*

-  (15)  *\[\[a×b\]×c\] +\[\[b×c\]×a\] +\[\[c×a\]×b\] =o*
-  (16)  *\[\[a×b\]×c\]=\[a×\[b×c\]\]+\[b×\[c×a\]\]*
+  (15)  *\[a×b\]×c + \[b×c\]×a + \[c×a\]×b = o*
+  (16)  *\[a×b\]×c = a×\[b×c\] + b×\[c×a\]*

Vectors modified by Vladimir Chukharev

Vladimir Chukharev — Mon, 12 Nov 2018 07:25:57 -0000

--- v42
+++ v43
@@ -42,7 +42,7 @@
   (12) *\[a×b\]×\[c×d\]=(a⋅\[b×d\])⋅c-(a⋅\[b×c\])⋅d=(a⋅\[c×d\])⋅b-(b⋅\[c×d\])⋅a*
   (13) *\[a×b\]×\[a×c\]=(a⋅\[b×c\])⋅a*

-General affine coordinates can determined according to the equation
+Components over a general linear coordinate basis a, b, c  can be determined according to the equation
   (14) *d=((d⋅\[b×c\])⋅a+(a⋅\[d×c\])⋅b+(a⋅\[b×d\])⋅c)/(a⋅\[b×c\])*
 where *a⋅\[b×c\]!=0*

Vectors modified by Vladimir Chukharev

Vladimir Chukharev — Sun, 28 Oct 2018 15:45:33 -0000

--- v41
+++ v42
@@ -27,7 +27,7 @@

 Also, the mixed product (or the scalar triple product), *a⋅(b×c)*, is the (signed) volume of the parallelepiped defined by the three vectors given. It is invariant under circular shift of the three parameters and changes sign under swapping of any two parameters.

-   (8) *a⋅\[b×c\] = \[a×b\]⋅c = \[b×c\]⋅a = b⋅\[c×a\] = \[c×a\]⋅b = c⋅\[a×b\] = -\[a×c\]⋅b = -a⋅\[c×b\]= -\[b×a\]⋅c = -b⋅\[a×c\] = -\[c×b\]⋅a = -c⋅\[b×a\] *
+   (8) *a⋅\[b×c\]=\[a×b\]⋅c=\[b×c\]⋅a=b⋅\[c×a\]=\[c×a\]⋅b=c⋅\[a×b\]=-\[a×c\]⋅b=-a⋅\[c×b\]=-\[b×a\]⋅c=-b⋅\[a×c\]=-\[c×b\]⋅a=-c⋅\[b×a\]*

 The rule "bac minus cab" allows to calculate triple cross-product using dot-products.

Vectors modified by Vladimir Chukharev

Vladimir Chukharev — Sun, 28 Oct 2018 15:12:46 -0000

--- v40
+++ v41
@@ -27,7 +27,7 @@

 Also, the mixed product (or the scalar triple product), *a⋅(b×c)*, is the (signed) volume of the parallelepiped defined by the three vectors given. It is invariant under circular shift of the three parameters and changes sign under swapping of any two parameters.

-   (8) *a⋅\[b×c\] = \[a×b\]⋅c = \[b×c\]⋅a = b⋅\[c×a\] = \[c×a\]⋅b = c⋅\[a×b\] = -\[a×c\]⋅b = -a⋅\[c×b\] = -\[c×b\]⋅a = -c⋅\[b×a\] = -\[b×a\]⋅c = -b⋅\[a×c\]*
+   (8) *a⋅\[b×c\] = \[a×b\]⋅c = \[b×c\]⋅a = b⋅\[c×a\] = \[c×a\]⋅b = c⋅\[a×b\] = -\[a×c\]⋅b = -a⋅\[c×b\]= -\[b×a\]⋅c = -b⋅\[a×c\] = -\[c×b\]⋅a = -c⋅\[b×a\] *

 The rule "bac minus cab" allows to calculate triple cross-product using dot-products.

Vectors modified by Vladimir Chukharev

Vladimir Chukharev — Sun, 28 Oct 2018 15:10:41 -0000

--- v39
+++ v40
@@ -27,7 +27,7 @@

 Also, the mixed product (or the scalar triple product), *a⋅(b×c)*, is the (signed) volume of the parallelepiped defined by the three vectors given. It is invariant under circular shift of the three parameters and changes sign under swapping of any two parameters.

-   (8) *a⋅\[b×c\] = \[a×b\]⋅c = \[b×c\]⋅a = \[c×a\]⋅b = -\[a×c\]⋅b = -\[c×b\]⋅a = -\[b×a\]⋅c = -a⋅\[c×b\]*
+   (8) *a⋅\[b×c\] = \[a×b\]⋅c = \[b×c\]⋅a = b⋅\[c×a\] = \[c×a\]⋅b = c⋅\[a×b\] = -\[a×c\]⋅b = -a⋅\[c×b\] = -\[c×b\]⋅a = -c⋅\[b×a\] = -\[b×a\]⋅c = -b⋅\[a×c\]*

 The rule "bac minus cab" allows to calculate triple cross-product using dot-products.

Vectors modified by Vladimir Chukharev

Vladimir Chukharev — Sun, 28 Oct 2018 15:03:38 -0000

--- v38
+++ v39
@@ -27,7 +27,7 @@

 Also, the mixed product (or the scalar triple product), *a⋅(b×c)*, is the (signed) volume of the parallelepiped defined by the three vectors given. It is invariant under circular shift of the three parameters and changes sign under swapping of any two parameters.

-   (8) *a⋅(b×c) = (a×b)⋅c = (b×c)⋅a = (c×a)⋅b = -(a×c)⋅b = -(c×b)⋅a = -(b×a)⋅c = -a⋅(c×b)*
+   (8) *a⋅\[b×c\] = \[a×b\]⋅c = \[b×c\]⋅a = \[c×a\]⋅b = -\[a×c\]⋅b = -\[c×b\]⋅a = -\[b×a\]⋅c = -a⋅\[c×b\]*

 The rule "bac minus cab" allows to calculate triple cross-product using dot-products.

Vectors modified by Vladimir Chukharev

Vladimir Chukharev — Sun, 28 Oct 2018 14:58:06 -0000

--- v37
+++ v38
@@ -47,5 +47,5 @@
 where *a⋅\[b×c\]!=0*

   (15)  *\[\[a×b\]×c\] +\[\[b×c\]×a\] +\[\[c×a\]×b\] =o*
-  (16) *\[\[a×b\]×c\] =\[a×\[b×c\]\] +\[b×\[c×a\]\] *
+  (16)  *\[\[a×b\]×c\]=\[a×\[b×c\]\]+\[b×\[c×a\]\]*

Vectors modified by Vladimir Chukharev

Vladimir Chukharev — Sun, 28 Oct 2018 14:56:22 -0000

--- v36
+++ v37
@@ -43,8 +43,9 @@
   (13) *\[a×b\]×\[a×c\]=(a⋅\[b×c\])⋅a*

 General affine coordinates can determined according to the equation
- (14) *d=((d⋅\[b×c\])⋅a+(a⋅\[d×c\])⋅b+(a⋅\[b×d\])⋅c)/(a⋅\[b×c\])*
- where *a⋅\[b×c\]!=0*
+  (14) *d=((d⋅\[b×c\])⋅a+(a⋅\[d×c\])⋅b+(a⋅\[b×d\])⋅c)/(a⋅\[b×c\])*
+where *a⋅\[b×c\]!=0*

   (15)  *\[\[a×b\]×c\] +\[\[b×c\]×a\] +\[\[c×a\]×b\] =o*
-  (15') *\[\[a×b\]×c\] =\[a×\[b×c\]\] +\[b×\[c×a\]\] *
+  (16) *\[\[a×b\]×c\] =\[a×\[b×c\]\] +\[b×\[c×a\]\] *
+