Scalar Triple Product

Introduction

Have you ever gotten stuck when trying to solve a scalar triple product problem? If you have, don’t worry. In this article, we will show you how to solve any scalar triple product problem. The answers to these problems are derived from the volume of the parallelepiped having every three vectors as its sides and called the physical significance of a.b.c triple product. Let us discuss in depth the scalar triple product formula for better understanding. The vectors’ scalar triple product is obvious from its name: it is the 3 vectors’ product. It entails multiplying dot products of one vector by the other two’s cross product. [a b c] = (a b) is how it’s written.

Looking at the aforementioned formula, the following outcome may be drawn:

i) A scalar quantity is always the outcome.

ii) In mathematics, the triple product is an extension of the double product, denoted by ⨂. It is a generalization of the dot product, the scalar triple product, and the vector triple product.

iii) The scalar triple product formula is one of the most curious and uncommon formulas in mathematics. A parallelepiped is a volume that has six equivalent faces, three of which are coterminous edges, like a cube or cube-shaped box, for example. The volume of any parallelepiped is computed by adding the volumes of its components (usually surfaces) in the same way that a sum can be made by adding three positive numbers. The scalar triple product formula, as discussed here, relates to the volume of one kind of parallelepiped: the one whose coterminous edges correspond to the three vectors a, b, and c.

Hence, it can be seen that [ a b c] = [ b c a ] = – [ a c b ] 

Proof of the Scalar Triple Product

abc = (a x b).c

A parallelepiped is a three-dimensional solid whose six faces are parallelograms each parallel to the bases. If a, b, and c are any three vectors in R3, then the parallelepiped determined by these vectors is the volume of the parallelepiped whose three coterminous edges imply these three vectors. The area of the base is determined by the cross product of two vectors (a and b) among these three sides. This result has a perpendicular direction to both vectors. The component of the third vector (say c) along the direction of the resulting cross product determines the height.

The parallelogram’s area is given by |a x b|, and this vector’s direction is upright to the base.

Further Explanation

Try to recollect determinants’ properties, as the notion of determinants aids in the easy solution of these issues.

iii) The vector set’s triple product is a result of multiplying the magnitudes of each vector, and the results are added together. If the final number equals zero, then the vectors are considered coplanar.

The parallelepiped volume is represented by the scalar triple product. If it’s zero, then such a situation could only occur if one of the three vectors has a magnitude of zero. The a & b’s cross product has a direction that is plane’s perpendicular that includes a and b. Only if c’s vector is also in the similar plane will the dot product of the resultant with c be zero. It is due to the angle being 90 and cos 90 between the C and resultant, this is the case.

If a, b, and c are three vectors, the scalar triple product (a * b) * c is represented as [a b c]. ∴ (a * b) * c = [a b c]

Geometrical Interpretation of Scalar Triple Product 

The parallelepiped volume has coterminous ends denoted by: a, b, and c. It makes the vectors’ right-handed framework.  And, we use scalar triple product to represent it as: (a * b) * c.

In the form of components, this expression is expressed as a = a1i + a1j + a1k, b = a2i + a2j + a2k, c = a3i +a3j + a3k. 

Properties of Scalar Triple Products. 

When the cyclic order is modified, there’s a change in the sign of the scalar triple product formula but its magnitude’s sign stays the same . [a b c] = – [a c b], and so on. In case two  scalar triple product’s vectors are the same, the product disappears. In other words, [a a b] = 0, [a b a] = 0, and [b an a] = 0. In case there are two collinear or parallel vectors, the scalar triple product disappears. [x a b c] Equals x [a b c] for every scalar x. Also, xyz [a b c] Equals [x a yb zc]. [a + b c d] = [a c d] + [b c d] for every vector a, b, c, d

Conclusion

The scalar triple product is an interesting geometric concept that provides us with a technique for calculating parallelepiped volume (which is some serious math magic). So, we’ll wrap this entry up with another interesting application of the scalar triple product. By the way, just to be clear, the scalar triple product is also referred to as the ‘dot product’ or the ‘dot operator’, depending on where you learnt your vector algebra from.