Scalar Triple Product

A dot product of a vector also with cross product of two additional vectors is called the scalar triple product. For example, if a, b, and c are three vectors, the scalar triple product is a .(b x c). The  scalar triple product, box product, & mixed product are other names for it. The volume of such a parallelepiped is calculated using the scalar triple product, where the 3 vectors indicate the parallelepiped’s neighbouring sides.

The notion of the scalar triple product and its formula, proof, and characteristics will be discussed in this article. We’ll also look at the geometric meaning of a scalar triple product & solve a few problems based on it to see how it’s used.

What is a Scalar Triple Product?

The scalar triple product of 3 vectors a, b, and c is equal to the scalar product between vector a and the cross product of vectors b and c, i.e., a .(bxc). It can alternatively be expressed symbolically as [a b c] = [a, b, c] = a · (b × c). The volume of a parallelepiped having adjacent sides a, b, and c is given by scalar triple product [a b c]. If three vectors a, b, and c are supplied, the scalar triple products [a b c] include:

•   a · (b × c)
•   a · (c × b)
•   b · (a × c)
•   b · (c × a)
•   c · (b × a)
•   c · (a × b)

Before we go into the formula for the scalar triple product, it’s important to remember that:

•   [a, b, c] = a · (b × c) = b · (c × a) = c · (a × b)
•   a · (b × c) = – a · (c × b)
•   b · (c × a) = – b · (a × c)
•   c · (a × b) = – c · (b × a)
•   a · (b × c) = (a × b) · c

Formula of Scalar Triple Product

If there are 3 vectors a = a1 i + a2 j + a3 k, b = b1 i + b2 j + b3 k, and c = c1 i + c2 j + c3 k, then the determinant of components of three vectors gives their scalar triple product.

A scalar triple product for vectors a, b, and c has the expression, 

[a,b,c]=|a1 a2 a3|

             |b1 b2 b3|

             |c1 c2 c3|

Scalar Triple Product Geometric Interpretation

We now know that the scalar triple product of whatever three vectors a, b, and c is a. (b× c), equivalent to the determinant of components of three vectors. Let’s look at the scalar triple product’s geometric meaning presently. The volume of a parallelepiped is given by the absolute amount of the scalar triple product a .(b× c), where a, b, and c are the parallelepiped’s neighbouring sides. The area of parallelogram produced by the vectors b c is given by the cross product (b × c). b × c is perpendicular to the surface containing vectors b and c, according to the definition cross product. 

Properties

We looked at the scalar triple product notion, as well as its geometrical interpretation & formula. For a better grasp of the notion, let us go through most of its key properties:

•   If any two of the three vectors are parallel, the scalar triple product is zero, i.e. [a a b] = 0.
•   [(a + b) c d] = [a c d] + [b c d]

LHS = [(a + b) c d]

= (a + b) · (c × d)

= a · (c × d) + b · (c × d)

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

= RHS

• [λa b c] = λ [a b c], where λ is a real number.

•   If and only if three non-zero vectors are coplanar, the scalar triple product is zero.
•   Commutative Property:

a · (b × c) = (b × c) · a

b · (c × a) = (c × a) · b

c · (a × b) = (a × b) · c 

Key Information about the Scalar Triple Product

• [a, b, c] = [b, c, a] = [c, a, b]
• [a (b+c) d] = [a b d] + [a c d], [a b (c+d)] = [a b c] + [a b d]
• [λa b c] = [a λb c] = [a b λc] = λ [a b c], where λ here is a real number
• If and only if three non-zero vectors are coplanar, the scalar triple product is zero

Conclusion 

The consequence of scalar triple products is always scalar quantities. Calculating the cross products of two vectors yields scalar triple product formulas. After that, compute a dot product of a leftover vector with the resulting vector.

If a triple product is 0, one of three vectors chosen is equivalent to zero magnitudes or the vectors are coplanar. Using this approach, the volume of a parallelepiped may be simply computed.