There are two types of triple product of vectors
(a) Scalar Triple Product : [katex](\underline u\times\underline v).\underline w[/katex] or [katex] \underline u.(\underline v \times \underline w)[/katex]
(b) Vector Triple Product : [katex](\underline u\times\underline v)\times\underline w[/katex]
In this section, we shall study the scalar triple product only
Definition : let [katex]\overrightarrow u[/katex] = [katex] a_1\underline i[/katex] + [katex] b_1\underline j [/katex] + [katex] c_1 \underline k[/katex] , [katex]\overrightarrow v[/katex] = [katex] a_2\underline i[/katex]+ [katex] b_2\underline j [/katex] + [katex] c_2 \underline k[/katex] , [katex]\overrightarrow w[/katex] = [katex] a_3\underline i[/katex] + [katex] b_3\underline j [/katex] + [katex] c_3 \underline k[/katex] The scalar triple product of vector [katex]\overrightarrow u[/katex] , [katex]\overrightarrow v[/katex] and [katex]\overrightarrow w[/katex] is defined by [katex]\overrightarrow u[/katex]. ( [katex]\overrightarrow v[/katex] × [katex]\overrightarrow w[/katex]) or [katex]\overrightarrow v[/katex].( [katex]\overrightarrow w[/katex] ×[katex]\overrightarrow u[/katex]) or [katex]\overrightarrow w[/katex].( [katex]\overrightarrow u[/katex] × [katex]\overrightarrow v[/katex]) The scalar triple product [katex]\overrightarrow u[/katex]. ( [katex]\overrightarrow v[/katex] × [katex]\overrightarrow w[/katex] [katex]\overrightarrow w[/katex]) is written as [katex]\overrightarrow u[/katex]. ( [katex]\overrightarrow v[/katex] × [katex]\overrightarrow w[/katex]) =[ [katex] \underline u\;\underline v\;\underline w [/katex] ]
Analytical Expression Of [katex]\overrightarrow u[/katex]. ( [katex]\overrightarrow v[/katex] [katex]\overrightarrow w[/katex] )
Let [katex]\overrightarrow u[/katex] = [katex] a_1\underline i[/katex] + [katex] b_1\underline j [/katex] + [katex] c_1 \underline k[/katex] , [katex]\overrightarrow v[/katex] = [katex] a_2\underline i[/katex]+ [katex] b_2\underline j [/katex] + [katex] c_2 \underline k[/katex] , [katex]\overrightarrow w[/katex] = [katex] a_3\underline i[/katex] + [katex] b_3\underline j [/katex] + [katex] c_3 \underline k[/katex]
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) ([katex]\overrightarrow u[/katex] × [katex]\overrightarrow v[/katex] ) [katex]\overrightarrow w[/katex] = [katex]\overrightarrow u[/katex] ([katex]\overrightarrow v[/katex] × [katex]\overrightarrow w[/katex]) = [ [katex] \underline u\;\underline v\;\underline w [/katex] ]
([katex]\overrightarrow v[/katex] × [katex]\overrightarrow w[/katex] ) [katex]\overrightarrow u[/katex] = [katex]\overrightarrow v[/katex] ([katex]\overrightarrow w[/katex] × [katex]\overrightarrow u[/katex]) = [ [katex] \underline v\;\underline w\;\underline u [/katex] ]
([katex]\overrightarrow w[/katex] × [katex]\overrightarrow u[/katex] ) [katex]\overrightarrow v[/katex] = [katex]\overrightarrow w[/katex] ([katex]\overrightarrow u[/katex] × [katex]\overrightarrow v[/katex]) = [ [katex] \underline w\;\underline u\;\underline v[/katex] ]
(3) The value of the product changes if the order is non cyclic .
(4) ([katex]\overrightarrow u[/katex]. [katex]\overrightarrow v[/katex] . [katex]\overrightarrow w[/katex] and [katex]\overrightarrow u[/katex] × ([katex]\overrightarrow v[/katex] . [katex]\overrightarrow w[/katex])
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 ([katex]\overrightarrow u[/katex] × [katex]\overrightarrow v[/katex] ) [katex]\overrightarrow w[/katex] represents the volume of the parallelepiped having [katex]\overrightarrow u[/katex] , [katex]\overrightarrow v[/katex] and [katex]\overrightarrow w[/katex] as its conterminous edges. As it is seen from the formula that:
([katex]\overrightarrow u[/katex] × [katex]\overrightarrow v[/katex] ) [katex]\overrightarrow w[/katex] = [katex] \left| \overrightarrow u × \overrightarrow v \right| [/katex] [katex] \left| \overrightarrow w \right| \cos\theta [/katex] Hence
(i) [katex] \left| \overrightarrow u × \overrightarrow v \right| [/katex] = area of the parallelogram with two adjacent sides, [katex]\overrightarrow u[/katex] and [katex]\overrightarrow v[/katex]
(ii) [katex] \left| \overrightarrow w \right| \cos\theta [/katex] = height of the parallelepiped
([katex]\overrightarrow u[/katex] × [katex]\overrightarrow v[/katex] ) [katex]\overrightarrow w[/katex] = [katex] \left| \overrightarrow u × \overrightarrow v \right| [/katex] [katex] \left| \overrightarrow w \right| \cos\theta [/katex] =(Area of parallelogram)(height)
= Volume of the parallelepiped Similarly, by taking the base plane formed by [katex]\overrightarrow v[/katex] and [katex]\overrightarrow w[/katex] , we have
The volume of the parallelepiped = ([katex]\overrightarrow v[/katex] × [katex]\overrightarrow w[/katex] ) [katex]\overrightarrow u[/katex]
And by taking the base plane formed by [katex] ] \overrightarrow w[/katex] and [katex] ] \overrightarrow u[/katex] , we have
The volume of the parallelepiped = ([katex]\overrightarrow w[/katex] × [katex]\overrightarrow u[/katex] ) [katex]\overrightarrow v[/katex]
So, we have: ([katex]\overrightarrow u[/katex] × [katex]\overrightarrow v[/katex] ) [katex]\overrightarrow w[/katex] = ([katex]\overrightarrow v[/katex] × [katex]\overrightarrow w[/katex] ) [katex]\overrightarrow u[/katex] = ([katex]\overrightarrow w[/katex] × [katex]\overrightarrow u[/katex] ) [katex]\overrightarrow v[/katex]
(2) The Volume Of The Tetrahedron :
Volume of the tetrahedron ABCD
=[katex]\frac13 [/katex] ( [katex] \triangle[/katex] ABC)(height of D above the place ABCD)
= [katex]\frac13 \frac12 \left| \overrightarrow u × \overrightarrow v \right| (h) [/katex]
= [katex]\frac16 [/katex] (Area of parallelogram with AB and AC as adjacent\)
= [katex]\frac16 [/katex] (volume of the parallelogram with [katex]\overrightarrow u[/katex] , [katex]\overrightarrow v[/katex] , [katex]\overrightarrow w[/katex] as edges)
Thus volume = [katex]\frac16 \overrightarrow u[/katex] × [katex]\overrightarrow v[/katex] ) [katex]\overrightarrow w[/katex] =[ [ [katex]\frac16 \underline u\;\underline v\;\underline w [/katex] ]
Properties of Triple scalar Product :
(1) If [katex]\overrightarrow u[/katex] , [katex]\overrightarrow v[/katex] and [katex]\overrightarrow w[/katex] are coplanar , then the volume of the parallelepiped so formed is zero i.e; the vectors [katex]\overrightarrow u[/katex] , [katex]\overrightarrow v[/katex] , [katex]\overrightarrow w[/katex] are coplanar [katex] \Leftrightarrow [/katex] ([katex]\overrightarrow u[/katex] × [katex]\overrightarrow v[/katex] ) [katex]\overrightarrow w[/katex] =0
(2) If any two vectors of triple product are equal , then its value is zero i.e;
[ [katex] \underline u\;\underline u\;\underline w [/katex] ] = [ [ [katex] \underline u\;\underline v\;\underline v [/katex] ]