Scalar Triple Product in Vector Algebra

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.