Scalar Triple Product: Volume and Coplanarity

1. Defining and Calculating the Scalar Triple Product

As its name suggests, the scalar triple product involves three vectors and results in a scalar value. It is defined as the dot product of one vector with the cross product of the other two.

For three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$, the scalar triple product is written as $\vec{a}\cdot (\vec{b}\times \vec{c})$. It is often denoted using a shorthand notation called the "box product": $[\vec{a}\vec{b}\vec{c}]$.

In our last lesson, you learned to compute the cross product first, then the dot product. However, there's a much more direct way to compute the scalar triple product using determinants, a topic you covered in Module 1. This method combines both steps into one.

The following video provides an excellent explanation of what the scalar triple product is and how this determinant "shortcut" works.

So, the key formula for calculation is:
If $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$, and $\vec{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$, then their scalar triple product is given by the 3x3 determinant of their components:

2. Geometric Interpretation: The Volume of a Parallelepiped

The scalar triple product isn't just an abstract calculation; it has a powerful geometric meaning. The absolute value of the scalar triple product, $|[\vec{a}\vec{b}\vec{c}]|$, represents the volume of the parallelepiped formed by the three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ when they are placed tail-to-tail (as coterminous edges).

Let's break down why this is true. The volume of any prism-like solid is (Area of Base) × (Height).

1. Area of the Base: As we learned, the magnitude of the cross product, $|\vec{b}\times \vec{c}|$, gives the area of the parallelogram forming the base.
2. Height: The vector $\vec{b}\times \vec{c}$ is perpendicular to the base. The dot product $\vec{a}\cdot (\vec{b}\times \vec{c})$ calculates the projection of $\vec{a}$ onto this perpendicular vector, effectively finding the height of the parallelepiped, and multiplies it by the base area.

The next segment of the JK Math video provides a detailed visual derivation of this relationship.

The formula for the volume is:

A related shape is the tetrahedron. The volume of a tetrahedron with coterminous edges $\vec{a}$, $\vec{b}$, and $\vec{c}$ is one-sixth of the volume of the corresponding parallelepiped.

This is a very common formula tested in the MHT-CET.

Let's watch a problem that uses these concepts in a typical MHT-CET context. This problem also involves finding a minimum value, which links to the "Application of Derivatives" topic.

3. Application: Testing for Coplanarity

What happens if the volume of the parallelepiped is zero? Geometrically, this means the three vectors cannot form a 3D solid; they must all lie flat on the same plane. Such vectors are called coplanar.

This gives us a crucial test:
Three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ are coplanar if and only if their scalar triple product is zero.

This concept is explained very clearly in the following short clip.

The condition for coplanarity is fundamental and is often used in problems where you need to find an unknown variable that makes three given vectors lie in the same plane.

Dinesh Sir also emphasizes this rule as it's a frequently tested concept in MHT-CET.

Now, let's practice this with a textbook example.

Conclusion

In this lesson, we've added a powerful tool to our vector toolkit. By combining the dot and cross products, we've unlocked the ability to solve problems in three dimensions.

Here are the key takeaways:

• The Scalar Triple Product (or box product) of vectors $\vec{a},\vec{b},\vec{c}$ is $[\vec{a}\vec{b}\vec{c}]=\vec{a}\cdot (\vec{b}\times \vec{c})$.
• It is most efficiently calculated using a 3x3 determinant of the vector components.
• The absolute value, $|[\vec{a}\vec{b}\vec{c}]|$, gives the volume of the parallelepiped with those vectors as coterminous edges.
• The scalar triple product is zero if and only if the three vectors are coplanar (lie in the same plane).

Mastering these two applications—volume and coplanarity—is essential for the MHT-CET. In our next lesson, we will shift our focus to representing lines in 3D space using both vector and Cartesian equations, further building our understanding of 3D geometry.