Scalar Triple Product

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Scalar Triple Product Formula

The product of three vectors in mathematics simply refers to the scalar triple product of vectors. The resultant vector is a scalar quantity and is represented as (a x b).c. In this formula, dot and cross can be interchanged, that is; (a x b).c = a.(b x c). Students can go through this article to learn more about the scalar triple product and vector triple product, its definition, formula, properties and more.


Proof of Scalar Triple Product

Scalar triple product formula means the dot product of one of the vectors with the cross product of the other two vectors. It can be written as:

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

The formula signifies the volume of the parallelepiped whose three coterminous edges denote three vectors, say, a, b and c. Among these three sides, the cross product of two vectors (let a and b) gives the area of the base. The direction of this result is perpendicular to both the vectors. The height is given by the component of the third vector (say c) along the direction of the resultant cross product. 

  • It means that |a x b| gives the area of the parallelogram and the direction of this vector 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. 

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From the above diagram, it can be shown that the direction of |a x b| vector is perpendicular to the base and |c|cos cos Ф denotes the height. 

From the definition of expansion of cross product of vectors, it becomes hassle-free to calculate the scalar triple product proof.

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\[_{3}\]\[\hat{k}\])

By using the determinants properties, scalar and vector triple product formula can be calculated:

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

Some of the conclusions can be drawn after taking a look at the formula of scalar triple product:

  • The resultant of a scalar triple product is always a scalar quantity.

  • To determine the formula for the scalar triple product, the cross product of two vectors is calculated first. After that, the dot product of the remaining vector with the resultant vector is calculated.

  • If the triple product results to be zero, then it suggests that one of the three vectors taken is of zero magnitudes. 

  • One can find out the volume of a parallelepiped effortlessly. 

Scalar Triple Product Properties

The scalar triple product is cyclic; that is;

[a b c] = [b c a] = [c a b] = -[b a c] = -[c b a] = -[a c b]

  • 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] = k[a b c]

  •  [(a + b) c d] = [(a + b) . (c + d)]

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

= [a c d] + [b c d]

Understand the formula of scalar triple product properly with a given example:


Question: 

Three Vectors are a = \[\hat{i}\] - \[\hat{j}\] + \[\hat{k}\], b = 2\[\hat{i}\] + \[\hat{j}\] + \[\hat{k}\], c = \[\hat{i}\] + \[\hat{j}\] - 2\[\hat{k}\]. Calculate the Scalar Triple Product and Verify if [a b c] = [b c a].


Answer: 

According to the scalar triple product: 

[a 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 b c] = ( a x b) . c = \[\begin{vmatrix} 1 & 1  & -2 \\ 1 & -1 & 1 \\ 2 & 1 & 1 \end{vmatrix}\] = -7

So, the answer comes out to be -7.

Now, to verify if [a b c] = [b c a], calculate [b c a] as

[b c a] = \[\begin{vmatrix} a_{1} & a_{2} & a_{3}\\  b_{1} & b_{2} & b_{3}\\  c_{1} & c_{2} & c_{3} \end{vmatrix}\] = -7

Hence, [a b c] = [b c a].

FAQ (Frequently Asked Questions)

Q1. Can Dot Product and Cross Product be Interchanged in the Scalar Triple Product?

The scalar triple product is denoted by (a x b).c. Here, dot product and cross product can be used interchangeably, without changing the order of vector occurrences. Using this interchangeability property, various properties of the scalar triple product can be derived:

  • Associative property: (a x b).c = a. (b x c)

  • Commutative property: (a x b).c = (b x c).a = (c x a).b

Moreover, if any two vectors taken in scalar and vector triple product are interchanged with respect to their position, then the value comes out to be (-1) of the original result:

[a b c] = [b c a] = [c a b] = -[a c b] = -[c b a] = -[b a c]