= a × b + a × c and this proves the distributive law for the cross
product.
Observation 14.5.1Suppose you have three vectors,u =
(a,b,c)
,v =
(d,e,f)
, andw =
(g,h,i)
. Thenu ⋅ v × wis given by the following.
| |
|| i j k || || e f || ||d f || || d e ||
u⋅v × w = (a,b,c)⋅||d e f || = a|| h i ||− b||g i ||+ c|| g h ||
|g h i|
( a b c )
= det( d e f ) .
g h i
The message is that to take the box product, you can simply take the determinant of thematrix which results by letting the rows be the rectangular components of the given vectorsin the order in which they occur in the box product.