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Mathematics Question on Vector Algebra

The value of [a⃗−b⃗      b⃗−c⃗      c⃗−a⃗][ \vec{a} - \vec{b} \,\,\,\,\,\, \vec{b} - \vec{c} \,\,\,\,\,\, \vec{c} - \vec{a} ] where ∣a⃗∣=1,∣b⃗∣=5,∣c⃗∣=3|\vec{a}| = 1 , |\vec{b} | = 5 , |\vec{c}| = 3

A

0

B

1

C

6

D

None of these

Answer

0

Explanation

Solution

[a⃗−b⃗      b⃗−c⃗      c⃗−a⃗]\left[\vec{a} -\vec{b} \,\,\,\,\,\, \vec{b} - \vec{c} \,\,\,\,\,\, \vec{c} - \vec{a}\right]
=(a⃗−b⃗).[(b⃗−c⃗)×(c⃗−a⃗)]= \left(\vec{a} -\vec{b}\right).\left[\left(\vec{b} - \vec{c} \right) \times\left(\vec{c} - \vec{a}\right)\right]
=(a⃗−b⃗).[b⃗×c⃗−b⃗×a⃗−c⃗×c⃗+c⃗×a⃗]= \left(\vec{a} -\vec{b}\right).\left[\vec{b}\times \vec{c} -\vec{b}\times \vec{a} - \vec{c}\times \vec{c} + \vec{c} \times \vec{a}\right]
=(a⃗−b⃗)[b⃗×c⃗−b⃗×a⃗+c⃗×a⃗]=\left(\vec{a} - \vec{b}\right) \left[\vec{b} \times \vec{c} - \vec{b} \times \vec{a} + \vec{c} \times \vec{a}\right]
=a⃗.(b⃗×c⃗)−a⃗.(b⃗×a⃗)+a⃗.(c⃗×a⃗)−b⃗.(b⃗×c⃗)+b⃗.(b⃗×a⃗)−b⃗.(c×a⃗)= \vec{a}.\left(\vec{b}\times \vec{c}\right) -\vec{a}.\left(\vec{b}\times \vec{a}\right)+\vec{a} .\left(\vec{c}\times \vec{a}\right) - \vec{b}.\left(\vec{b}\times \vec{c}\right) + \vec{b}. \left(\vec{b}\times \vec{a}\right)- \vec{b}.\left(c\times \vec{a}\right)
=[a⃗   b⃗   c⃗]−[a⃗   b⃗   a⃗]+[a⃗   c⃗   a⃗]−[b⃗   b⃗   c⃗]+[b⃗   b⃗   a⃗]−[b⃗   c⃗   a⃗]= \left[\vec{a} \,\,\, \vec{b} \,\,\, \vec{c}\right] - \left[\vec{a} \,\,\, \vec{b} \,\,\, \vec{a}\right] + \left[\vec{a} \,\,\, \vec{c} \,\,\, \vec{a}\right] -\left[\vec{b} \,\,\, \vec{b} \,\,\, \vec{c}\right] + \left[\vec{b} \,\,\, \vec{b} \,\,\, \vec{a} \right] - \left[\vec{b} \,\,\, \vec{c} \,\,\, \vec{a} \right]
=[a⃗   b⃗   c⃗]−[b⃗   c⃗   a⃗]=0= \left[\vec{a} \,\,\, \vec{b} \,\,\, \vec{c} \right] - \left[\vec{b} \,\,\, \vec{c} \,\,\, \vec{a} \right] = 0

Three vectors are combined to form the scalar triple product of vectors. Because it evaluates to a single integer, much like the dot product, it is a scalar product. It requires multiplying the dot product of one of the vectors by the cross product of the other two vectors. It is expressed mathematically as (a x b)xc. The scalar triple product can be used to indicate the volume of a parallelepiped. The dot product of one vector and the cross product of the other two vectors is the scalar triple product, sometimes referred to as the mixed product, box product, or triple scalar product.

When vectors are multiplied and a scalar triple product is formed, the formula for the triple product of three vectors, or the dot product of a vector with the cross product of the other two vectors, is known as the scalar triple product formula. It is written as follows:

[a b c] = (a x b).c

In this instance (a x b). c is a scalar quantity and the resultant vector. This formula can alternatively be expressed by swapping the cross and dot in the middle, as seen below (a x b). c is the same as a. (b x c). The scalar triple product of vectors is obvious from its name: it is the product of three vectors. It requires multiplying one of the vectors by the cross product of the other two.

The properties of scalar triple product of vectors are:

  • If the vectors are permuted cyclically, then (a × b) . c = a.( b × c)
  • If the vectors are permuted cyclically, then a.(b × c) = b.(c × a) = c.(a × b)
  • If the triple product of vectors is zero, it can be assumed that the vectors are coplanar.