Scalar Triple Product

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Vector is said to be a quantity that helps to represent through forms of magnitude and direction. This vector may either look like an arrow which may point towards any direction of object.

The scalar triple product of three vectors a, b and c is represented as (a × b) ⋅c. This is said to be a scalar product, which is similar to that of dot product and helps to evaluate to single number. The scalar product turns out to be an important aspect as it comes with absolute value | (a × b) ⋅c| which is said to be the volume of parallelepiped spanned by a, b and c.

The triple product is known to be an important concept when it comes to vector calculus. This product has three vectors and is defined in three dimensions. There are two different types of triple products:

  1. Scalar triple product
  2. Vector triple product

Properties of scalar triple product

  1. The position of operators can be shifted without changing the order of operands. Therefore, it helps to keep the value of product:

x⃗.(y⃗ x z⃗) = (x⃗ x y⃗). z⃗

  1. The value of scalar triple product does not change in case the operands undergo any circular shift:

x⃗.(y⃗ x z⃗) = y⃗. (z⃗ x x⃗) = z⃗. (x⃗ x y⃗)

  1. While swapping two vectors, the scalar triple product is negative

x⃗. (y⃗x z⃗ ) = – x⃗. (z⃗ x y⃗)

Scalar Triple Product 1

  1. The scalar triple product turns out to be equal to that of determinant of components of any vector that is involved

Scalar Triple Product 2

a⃗.(b⃗ ×c⃗ )= ax  ayaz

bx bybz

cx   cycz


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