There are two types of triple product of vectors
(a) Scalar Triple Product : (\underline u\times\underline v).\underline w or \underline u.(\underline v \times \underline w)
(b) Vector Triple Product : (\underline u\times\underline v)\times\underline w
In this section, we shall study the scalar triple product only
Definition : let \overrightarrow u = a_1\underline i + b_1\underline j + c_1 \underline k , \overrightarrow v = a_2\underline i+ b_2\underline j + c_2 \underline k , \overrightarrow w = a_3\underline i + b_3\underline j + c_3 \underline k The scalar triple product of vector \overrightarrow u , \overrightarrow v and \overrightarrow w is defined by \overrightarrow u. ( \overrightarrow v × \overrightarrow w) or \overrightarrow v.( \overrightarrow w ×\overrightarrow u) or \overrightarrow w.( \overrightarrow u × \overrightarrow v) The scalar triple product \overrightarrow u. ( \overrightarrow v × \overrightarrow w \overrightarrow w) is written as \overrightarrow u. ( \overrightarrow v × \overrightarrow w) =[ \underline u\;\underline v\;\underline w ]
Analytical Expression Of \overrightarrow u. ( \overrightarrow v \overrightarrow w )
Let \overrightarrow u = a_1\underline i + b_1\underline j + c_1 \underline k , \overrightarrow v = a_2\underline i+ b_2\underline j + c_2 \underline k , \overrightarrow w = a_3\underline i + b_3\underline j + c_3 \underline k
Important Note : (1) The value of the triple scalar product depends upon the cycle order of the vectors , but is independent of the dot and cross. So the dot and cross , may be interchanged without alternating the value i.e; (2) (\overrightarrow u × \overrightarrow v ) \overrightarrow w = \overrightarrow u (\overrightarrow v × \overrightarrow w) = [ \underline u\;\underline v\;\underline w ]
(\overrightarrow v × \overrightarrow w ) \overrightarrow u = \overrightarrow v (\overrightarrow w × \overrightarrow u) = [ \underline v\;\underline w\;\underline u ]
(\overrightarrow w × \overrightarrow u ) \overrightarrow v = \overrightarrow w (\overrightarrow u × \overrightarrow v) = [ \underline w\;\underline u\;\underline v ]
(3) The value of the product changes if the order is non cyclic .
(4) (\overrightarrow u. \overrightarrow v . \overrightarrow w and \overrightarrow u × (\overrightarrow v . \overrightarrow w)
Applications Of Scalar Triple Product :
- The Volume Of The Parallelepiped
- The Volume Of The Tetrahedron
(1) The Volume Of The Parallelepiped :
The triple scalar product (\overrightarrow u × \overrightarrow v ) \overrightarrow w represents the volume of the parallelepiped having \overrightarrow u , \overrightarrow v and \overrightarrow w as its conterminous edges. As it is seen from the formula that:
(\overrightarrow u × \overrightarrow v ) \overrightarrow w = \left| \overrightarrow u × \overrightarrow v \right| \left| \overrightarrow w \right| \cos\theta Hence
(i) \left| \overrightarrow u × \overrightarrow v \right| = area of the parallelogram with two adjacent sides, \overrightarrow u and \overrightarrow v
(ii) \left| \overrightarrow w \right| \cos\theta = height of the parallelepiped
(\overrightarrow u × \overrightarrow v ) \overrightarrow w = \left| \overrightarrow u × \overrightarrow v \right| \left| \overrightarrow w \right| \cos\theta =(Area of parallelogram)(height)
= Volume of the parallelepiped Similarly, by taking the base plane formed by \overrightarrow v and \overrightarrow w , we have
The volume of the parallelepiped = (\overrightarrow v × \overrightarrow w ) \overrightarrow u
And by taking the base plane formed by ] \overrightarrow w and ] \overrightarrow u , we have
The volume of the parallelepiped = (\overrightarrow w × \overrightarrow u ) \overrightarrow v
So, we have: (\overrightarrow u × \overrightarrow v ) \overrightarrow w = (\overrightarrow v × \overrightarrow w ) \overrightarrow u = (\overrightarrow w × \overrightarrow u ) \overrightarrow v
(2) The Volume Of The Tetrahedron :
Volume of the tetrahedron ABCD
=\frac13 ( \triangle ABC)(height of D above the place ABCD)
= \frac13 \frac12 \left| \overrightarrow u × \overrightarrow v \right| (h)
= \frac16 (Area of parallelogram with AB and AC as adjacent\)
= \frac16 (volume of the parallelogram with \overrightarrow u , \overrightarrow v , \overrightarrow w as edges)
Thus volume = \frac16 \overrightarrow u × \overrightarrow v ) \overrightarrow w =[ [ \frac16 \underline u\;\underline v\;\underline w ]
Properties of Triple scalar Product :
(1) If \overrightarrow u , \overrightarrow v and \overrightarrow w are coplanar , then the volume of the parallelepiped so formed is zero i.e; the vectors \overrightarrow u , \overrightarrow v , \overrightarrow w are coplanar \Leftrightarrow (\overrightarrow u × \overrightarrow v ) \overrightarrow w =0
(2) If any two vectors of triple product are equal , then its value is zero i.e;
[ \underline u\;\underline u\;\underline w ] = [ [ \underline u\;\underline v\;\underline v ]