Scalar Triple Product

We can define the scalar triple product as the cross product of 2 vectors with a dot product of 1 given vector. That is, suppose a, b, c, are three given vector values, then we can denote the formula for the scalar triple product for these three vectors as:

a · (b × c)  

Some other names for scalar triple product are mixed product and box product. It is used to determine the value of a parallelepiped’s volume where the given 3 vectors (a,b,c) denote a parallelepiped’s adjacent sides. 

We can draw the outcomes below based on the formula mentioned above:

1. The product is always a scalar quantity.
2. In mathematics, the triple product (⨂) is an extension of the double product. It is a generalisation of the dot product, the scalar triple product, and the vector triple product.
3. The scalar triple product formula is one of the most 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. 

We can compute the volume of any parallelepiped by adding the volumes of its components (usually surfaces) in the same way we obtain a sum by adding three positive numbers. 

The scalar triple product formula, as discussed here, relates to the volume of a particular kind of parallelepiped. In this parallelepiped, the 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 ] 

Scalar Triple Product in More Details

As we know, the scalar triple product of 3 given vectors, say,  a, b, c, refers to the vector a’s scalar product with the vector b and c’s cross product. We can express it as:

a · (b × c). 

OR

[a b c] = [a, b, c] = a · (b × c)

Using the formula, we find the volume of the parallelepiped with adjacent sides denoted by vectors a, b, and c. 

Suppose, a, b, c, are 3 given vectors then the expressions for scalar triple products of a, b,and c will be:

• a*(bxc)
• a*(cxb)
• b*(axc)
• b*(cxa)
• C*(bxa)
• C*(axb)

Before we move to scalar triple product formula, note that:

• [s; b. c] = a (bx c) = b (cxa)=c-(axb)
• a (bxc)=-a (exb)
• b-(cxa)=-b-(ac)
• c-(axb)=-c (bxa)
• a-(bxc) = (axb)-c

Inverse Scalar Triple Product Formula

Denominating the three vectors A = A1 I + A2J + A3K; B = B1 I + B2 J + B3 K; and C = C1 I+ C2 J+ C3 K yields the scalar triple product. 

Scalar Triple Product Demonstration

In mathematics, the dot product of one of the vectors and the cross product of the other two vectors is called the scalar triple product formula. It is also possible to write it this way:

abc = (a x b).c

a, b, and c are three vectors designated by the parallelepiped’s three coterminous edges. The formula represents the parallelepiped’s volume. 

We can calculate the area of the base by taking the cross-product of two vectors on each of these three sides (let a and b). This result has a direction perpendicular to the two vectors in the issue. For instance, For instance, the third vector’s (say c) component determines its height in the direction of the resulting cross product.

As a result, the vector |a x b| gives the parallelogram’s area, with the direction perpendicular to the base. |c| cos indicates the height, representing the angle formed by a x b and c; a and b form the angle c.

Scalar Triple Product Characteristics

Below are the characteristics of scalar triple products.

• The nature of the scalar triple product is cyclic;

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

• We can define the scalar triple product using cyclically permuted vectors (a,b,c) using the following formula:

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

• If the scalar triple product of the given three vectors has the output as zero, all three vectors are known to be coplanar. The following expression represents it: 

ka kb kc = kabc

Where k can be any real number. 

• (a+b)cd = (a+b).(c+d)= a.(cxd)+b.(cxd)= acd + bcd

Conclusion

Here we thoroughly learned about scalar triple products. The article has covered all parts, from a well-explained description to formula and characteristics. 

The questions from the topic are likely to come in multiple examinations. So, make sure to understand the theory and concept better. Ensure to solve scalar triple product examples questions as much as possible to grasp the idea better.