triple product remains the same. ( ) If any two vectors are interchanged in their position in a scalar triple product, then the value of the scalar triple product is ( ) times the original value. More explicitly, [ , , ] a b c = [ , , ] [ , , ] [ , , ] [ , , ] [ , , ] b c a c a b a c b c b a b a c . Vector - - Theorem .
The scalar triple product preserves addition and scalar multiplication. That is, [( ), , ] b c d = [ , , ] [ , , ] a c d b c d ; , , ] a b c = [ , , ], a b c ∀∈ [ ,( ), ] a b c d = [ , , ] [ , , ] a b d a c d ; [ , , ] b c = [ , , ], a b c ∀∈ [ , ,( )] a b c = [ , , ] [ , , ] a b c a b d ; [ , , ] a b = [ , , ], a b c ∀∈ . Proof Using the properties of scalar product and vector product, we get [( ), , ] b c d = (( = ( = ( = [ , , ] [ , , ] a c d b c d , , ] a b c = (( ( ( )) (( [ , , ] a b c . Using the first statement of this result, we get the following.
[ ,( ), ] a b c d = [( ), , ] [ , , ] [ , , ] c d a b d a c d a = [ , , ] [ , , ] a b d a c d [ , , ] b c