Scalar Triple Product

In mathematics, the product of three vectors refers to the scalar triple product of vectors. Scalar quantities are derived from this formula and are expressed as (a x b).c. The dot and cross in this formula can be interchanged, that is, (a x b).c = a.(b x c). The purpose of this article is to teach students about the definition, formula, properties and more of the scalar triple product and vector triple product.

A Proof of Scalar Triple Products

Using a scalar triple product formula, we combine the cross product of two of the vectors and the dot product of one of the vectors. We can write it as follows:

abc= (a x b).c

This formula indicates the volume of a parallelepiped with three coterminous edges, for example, a, b, and c. In terms of the volume, the cross product of two vectors (let a and b be the vectors) results in the volume of the base. As a result, we get a perpendicular direction to both vectors. By calculating the height along the direction of the resultant cross product we can find the third vector (say c). 

• In other words, the parallelogram's area is a product of |a x b|, and the direction the vector faces is perpendicular to the base.

• The height is denoted by |c| cos cos Ф, where Ф denotes the angle between a x b and c. 

• The diagram above shows that the direction of the vector |a x b| is perpendicular to the base of the diagram, and denotes the height as |c| cos cos Ф.

• By defining the expansion of vector cross products, calculating the scalar triple product proof becomes a breeze.

Let a= a\[_{1}\]\[\hat{i}\] + a\[_{2}\]\[\hat{j}\] + a\[_{3}\]\[\hat{k}\], b= b\[_{1}\]\[\hat{i}\] + b\[_{2}\]\[\hat{j}\] + b\[_{3}\]\[\hat{k}\], c = c\[_{1}\]\[\hat{i}\] + c\[_{2}\]\[\hat{j}\] + c\[_{3}\]\[\hat{k}\]

Now, (a x b) . c = \[\begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\ a_{1} & a_{2} & a_{3}\\ b_{1} & b_{2} & b_{3} \end{vmatrix}\]. (c\[_{1}\] \[\hat{i}\] + c\[_{2}\]\[\hat{j}\] + c\[_{3}\]\[\hat{k}\])

(a x b). c =  \[\begin{vmatrix}\hat{i}.(c_{1} \hat{i} + c_{2}\hat{j} + c_{3}\hat{k}) & \hat{j}. (c_{1} \hat{i} + c_{2}\hat{j} + c_{3}\hat{k}) & \hat{k}.(c_{1} \hat{i} + c_{2}\hat{j} + c_{3}\hat{k})\\ a_{1} & a_{2} & a_{3}\\ b_{1} & b_{2} & b_{3} \end{vmatrix}\]

From the properties of the dot product of vectors:

\[\hat{i}\]. \[\hat{i}\] = \[\hat{j}\]. \[\hat{j}\] = \[\hat{k}\]. \[\hat{k}\] = 1 (cos 0 = 1)

It implies \[\hat{i}\]. (c\[_{1}\] \[\hat{i}\] + c\[_{2}\]\[\hat{j}\] + c\[_{3}\]\[\hat{k}\]) = c\[_{1}\]

\[\hat{j}\]. (c\[_{1}\] \[\hat{i}\] + c\[_{2}\]\[\hat{j}\] + c\[_{3}\]\[\hat{k}\]) = c\[_{2}\]

\[\hat{k}\]. (c\[_{1}\] \[\hat{i}\] + c\[_{2} \hat{j}\] + c\[_{3}\]\[\hat{k}\]) = c\[_{3}\]

 (a x b) . c = \[\begin{vmatrix} c_{1} & c_{2} & c_{3}\\ a_{1} & a_{2} & a_{3}\\ b_{1} & b_{2} & b_{3} \end{vmatrix}\]

(a x b) . c = \[\begin{vmatrix} c_{1} & c_{2} & c_{3}\\ a_{1} & a_{2} & a_{3}\\ b_{1} & b_{2} & b_{3} \end{vmatrix}\]

Scalar Triple Product Properties

The scalar triple product is cyclic; that is;

abc = bca = cab = -bac = -cba = -acb

• If the vectors taken in scalar triple product definition, say a, b, and c are cyclically permuted, then:

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

• If the scalar triple product of three vectors comes out to be zero, then it shows that given vectors are coplanar.

For any k that belongs to Real number,

Ka kb kc = kabc

•  (a+b)cd = (a+b).(c+d)

= a.(cxd)+b.(cxd)

= acd + bcd

Analysing the scalar triple product formula, some conclusions can be drawn:

• Scalar triple products always produce scalar quantities as their resultant.

• Scalar triple product formulas are determined by calculating cross products of two vectors. Thereafter, the dot product of the remaining vector and the resultant vector is calculated.

• This suggests one of the three vectors taken is equal to zero magnitudes if the triple product is zero. 

• A parallelepiped can be easily calculated by using this method.