Vector Triple Product :
`text(Definition:)` The vector triple product of three vectors `vec(a),vec(b),vec(c)` is defined as the vector product of two vectors `vec(a)` and `vec(b)xxvec(c)` .It is denoted by `vec(a) xx (vec(b)xxvec(c))`.
` (vec(a)xxvec(b) ) xxvec(c)` is a vector which is coplanar with `vec(a)` and `vec(b)` and perpendicular to `vec(c)`.
Hence ` ( vec(a) xx vec(b)) xx vec(c) = x vec(a) + yvec(b)` ......................(1) [linear combination of `vec(a)` and `vec(b)`)
`vec(c) * (vec(a) xx vec(b)) xxvec(c) = x (vec(a)*vec(c)) +y (vec(b)*vec(c))`
`0 = x (vec(a)*vec(c)) + y (vec(b)*vec(c))` ........................(2)
`:.` `x/ (vec(b)*vec(c)) = - y /(vec(a)*vec(c)) = lambda`
`:.` `x = lambda (vec(b)*vec(c)) ` and `y = -lambda (vec(a)*vec(c))`
Substituting the values of `x` and `y` in `(vec(a) xx vec(b)) xx vec(c) = lambda (vec(b)*vec(c)) vec(a) - lambda (vec(b)*vec(c)) vec(b)`
This is an identity and must be true for all values of `vec(a),vec(b),vec(c)`
Put = `vec(a) = hat(i) ; vec(b )= hat(j) ` and ` vec(c) = hat(i)`
`( hat(i) xx hat(j) )xx hat(i) = lambda (hat(j)xx hat(i)) hat(i) - lambda (hat(i) xx hat(i)) hat(j)`
`hat(j) = - lambda hat(j) => lambda =- 1`
Hence `(vec(a) xx vec(b)) xx vec(c ) = ( vec(a)*vec(c)) vec(b) - (vec(b)*vec(c)) vec(a) `
`text(Properties :)`
`1.` Expansion formula for vector triple product is given by
`quadquadquadquadvec(a) xx (vec(b) xx vec(c)) = (vec(a)*vec(c)) vec(b) - (vec(a)*vec(b)) vec(c)`
` (vec(b) xx vec(c)) xx vec(a) = (vec(b)*vec(a)) vec(c) - (vec(c)*vec(a))vec(b)`
`2.` ` [ (vec(a) xx vec(b) , vec(b) xx vec(c), vec(c) xx vec(a) )] = [ vec(a)vec(b)vec(c)]^2 = | (vec(a)*vec(a) , vec(a)*vec(b) , vec(a)*vec(c) ) , (vec(b)*vec(a) , vec(b)*vec(b) , vec(b)*vec(c) ) ,( vec(c)*vec(a) , vec(c)*vec(b) ,vec(c)*vec(c)) |`
Note that if `vec(a), vec(b) ,vec(c)` are non coplanar vectors then `vec(a) xx vec(b) ,vec(b) xx vec(c)` and `vec(c) xx vec(a)` will also be non coplanar vectors.
`3.` Vector triple product is a vector quantity.
`4.` ` vec(a) xx (vec(b) xx vec(c) ) ne (vec(a) xx vec(b)) xx vec(c) `.
`5.` Unit vector coplanar with `vec(a) ` & `vec(b)` and perpendicular to `vec(c) ` is `pm ( (vec(a) xx vec(b)) xx vec(c) ) / (| (vec(a) xx vec(b)) xx vec(c) | )`
` (vec(a)xxvec(b) ) xxvec(c)` is a vector which is coplanar with `vec(a)` and `vec(b)` and perpendicular to `vec(c)`.
Hence ` ( vec(a) xx vec(b)) xx vec(c) = x vec(a) + yvec(b)` ......................(1) [linear combination of `vec(a)` and `vec(b)`)
`vec(c) * (vec(a) xx vec(b)) xxvec(c) = x (vec(a)*vec(c)) +y (vec(b)*vec(c))`
`0 = x (vec(a)*vec(c)) + y (vec(b)*vec(c))` ........................(2)
`:.` `x/ (vec(b)*vec(c)) = - y /(vec(a)*vec(c)) = lambda`
`:.` `x = lambda (vec(b)*vec(c)) ` and `y = -lambda (vec(a)*vec(c))`
Substituting the values of `x` and `y` in `(vec(a) xx vec(b)) xx vec(c) = lambda (vec(b)*vec(c)) vec(a) - lambda (vec(b)*vec(c)) vec(b)`
This is an identity and must be true for all values of `vec(a),vec(b),vec(c)`
Put = `vec(a) = hat(i) ; vec(b )= hat(j) ` and ` vec(c) = hat(i)`
`( hat(i) xx hat(j) )xx hat(i) = lambda (hat(j)xx hat(i)) hat(i) - lambda (hat(i) xx hat(i)) hat(j)`
`hat(j) = - lambda hat(j) => lambda =- 1`
Hence `(vec(a) xx vec(b)) xx vec(c ) = ( vec(a)*vec(c)) vec(b) - (vec(b)*vec(c)) vec(a) `
`text(Properties :)`
`1.` Expansion formula for vector triple product is given by
`quadquadquadquadvec(a) xx (vec(b) xx vec(c)) = (vec(a)*vec(c)) vec(b) - (vec(a)*vec(b)) vec(c)`
` (vec(b) xx vec(c)) xx vec(a) = (vec(b)*vec(a)) vec(c) - (vec(c)*vec(a))vec(b)`
`2.` ` [ (vec(a) xx vec(b) , vec(b) xx vec(c), vec(c) xx vec(a) )] = [ vec(a)vec(b)vec(c)]^2 = | (vec(a)*vec(a) , vec(a)*vec(b) , vec(a)*vec(c) ) , (vec(b)*vec(a) , vec(b)*vec(b) , vec(b)*vec(c) ) ,( vec(c)*vec(a) , vec(c)*vec(b) ,vec(c)*vec(c)) |`
Note that if `vec(a), vec(b) ,vec(c)` are non coplanar vectors then `vec(a) xx vec(b) ,vec(b) xx vec(c)` and `vec(c) xx vec(a)` will also be non coplanar vectors.
`3.` Vector triple product is a vector quantity.
`4.` ` vec(a) xx (vec(b) xx vec(c) ) ne (vec(a) xx vec(b)) xx vec(c) `.
`5.` Unit vector coplanar with `vec(a) ` & `vec(b)` and perpendicular to `vec(c) ` is `pm ( (vec(a) xx vec(b)) xx vec(c) ) / (| (vec(a) xx vec(b)) xx vec(c) | )`