

Examples 
Ex 1: What is the volume of the parallelpiped spanned by the three vectors (– 2, 3, 1), (0, 4, 0) and (– 1, 3, 3) ?
Sol: Let = – 2 + 3 +
= 4 = – + 3 + 3 The required volume(V) is given by the absolute value of the scalar triple product.
( × ). 
= 


= 
– (12 – 0) – 3(0 – 0) + 1(0 + 4) 

= 
– 24 – 0 + 4 = – 20 
∴ V 
= 
– 20 = 20 

Ex 2: Find the volume of the tetrahedran having the edges i + j + k, i – j and i + 2j + k.
Sol: Let a = i + j + k, b = i – j
and c = i + 2j + k
⇒ [a b c] 
= 


= 
1(– 1) – 1(1) + 1(2 + 1) = 1 
∴ The volume of tetrahedran is
 [a b c]
= 1 = 
Ex 3: If = 2 – + , = – 3 – 5 and = 3 – 4 – 4, then find ( × ).

Sol: ( X ). 
= 
[ ] 

= 
= 0 
In this case, vector is perpendicular to ( X ) 

If , , are 3 vectors, ( × ) . is defined as the scalar product of the three vectors.
It is usually denoted by [ ]
Let OA = , OB = and OC = represent 3 noncoplanar vectors.
We can have a parallelopiped OADBFCGE with OA, OB and OC as its conterminous edges. Refer adjacent figure.
Let its volume be V, then
( × ) . = V
Note that for a lefthanded system, the volume being the same
( × ) . = – V
When does the scalar triple product equates to zero ?
Obviously ( × ) . = 0 under these 3 conditions:
i) any one (or more) of the 3 vectors is zero vector
ii) any two vectors are collinear
iii) vector is perpendicular to ( × )
For any 3 vectors , and we have the following:
i) ( × ) . = ( × ) . = ( × ) .
i.e, [ ] = [ ] = [ ]
ii) ( × ) . = . ( × )
i.e, the dot and cross operations can be swapped (interchanged).
iii) If no two vectors are collinear, then
( × ) . = 0 iff , , are coplanar
iv) Let
= a_{1} + a_{2} + a_{3};
= b_{1} + b_{2} + b_{3} ;
= c_{1} + c_{2} + c_{3}
(v) Say, any four points A, B, C, D are coplanar.
⇔ [AB AC AD] = 0
or [BA BC BD] = 0
or [CA CB CD] = 0
or [DA DB DC] = 0
(vi) The volume of the tetrahedron with a, b and c as coterminus edges is
[a b c]
(vii) The volume of the tetrahedron whose vertices are A, B, C, D is
 [AB AC AD] 
Note: Since Dot product is not defined between a scalar and a vector
( . ) . and . ( . ) do not convey any meaning and are nonexistent.