Proof
Let m =
X
∴ (
X
) X (
X
)
=
(
X
) X m
=
(
. m)
– (
. m)
=
(
. (
X
))
– (
. (
X
))
=
[
]
– [
]
Let
X
= n
Then (
X
) X (
X
)
=
n X (
X
)
=
(n .
)
– (n .
)
=
((
X
) .
)
– ((
X
) .
)
=
[
]
– [
]
Thus [
]
– [
]
= (
X
) X (
X
) = [
]
– [
]
We have proved that, if
,
,
are any there non-coplanar vectors and
is any vector, then
= x
+ y
+ z
where (x, y, z) is a unique triad of scalars. In the following theorem, we express x, y and z in terms of scalar triple products.
X 
X 
) X ( 
is any vector, then