Scalar Triple Product - JEE PDF Download

The scalar triple product is a significant concept in vector algebra that involves multiplying three vectors together. This operation can be accomplished by taking the dot product of one vector with the cross product of the other two vectors, resulting in a scalar quantity. The dot product always yields a specific value.
The scalar triple product is alternatively referred to as the box product, the triple scalar product, and the mixed product. In this section, individuals will acquire knowledge about the geometric interpretation of the scalar triple product, its formula and expansion, properties, and more.

Definition

If  and  are three vectors, then the scalar triple product is defined as the dot product of  with the cross product  and  i.e. dot product of with
It is generally denoted by

or 

Scalar triple product is read as box product of

Geometrical Interpretation of Scalar Triple Product

We can find the volume of a parallelopiped using the scalar triple product.

Area of parallelogram is cross product of vectors  and  i.e. 

Height of parallelopiped is
Then, the volume of parallelopiped is

So, we can say that 

is the volume of parallelopiped with coterminous edges   and

Formula

The scalar triple product equation is given as

Where

Properties

Below are some of the important properties of the scalar triple product.

• If we interchange the position of (.) and (x), the result will be the same, i.e.,
  
• Value of the scalar triple product doesn’t change if we don’t change the cyclic order of
  
• If we change the cyclic order of the vectors, then the sign of the scalar triple product is changed. i.e.,
  
  If any two of three vectors are equal or parallel, then the scalar triple product is zero.
  
• If three vectors are mutually perpendicular, then the scalar triple product is ±1.
  
• If three vectors are coplanar, then  This is the necessary and sufficient condition for three non zero and non-collinear vectors to be coplanar. 
• For any four vectors  and
  
• 
•  is always zero.
  
  Where 
  
• 

Scalar Triple Product Proof

If


Expansion: 

Finding Volume of Tetrahedron

Let and are the position vectors of vertices
A, B, C with respect to O, then the volume of tetrahedron OABC is given by 

Volume = ⅓ (area of base) x height
Area of base =  
Let

Area of base =
Height = projection of  

Volume =
Volume =