In this explainer, we will learn how to calculate the scalar triple product and apply this in geometrical applications.
Before looking at the scalar triple product, you should already be familiar with the scalar product (dot product) and the cross product. Here, we introduce a product of three vectors, combining both scalar and cross products.
Definition: Scalar Triple Product
The scalar triple product of three vectors , , and is defined as
We can write the scalar triple product as as well. However, the parentheses are unnecessary since carrying out first yields a scalar, and we cannot perform a cross product between a scalar and a vector.
The scalar triple product yields a scalar, as suggested by its name.
We know that
Property: Calculating a Scalar Triple Product Using the Vectors’ Components
Calculating the scalar triple product is equivalent to calculating the determinant
Let us apply this with a first example.
Example 1: Calculating the Scalar Triple Product of Three Vectors
Given , , and , find .
We know that calculating is equivalent to calculating . So, let us substitute the components of vectors , , and to calculate this determinant:
Let us look now at some properties of the scalar triple product. We know that swapping two horizontal rows in a determinant changes its sign. Therefore, switching two vectors in the scalar triple product changes the sign of the product:
If we perform a further permutation, it will reverse the sign again. So, by permutating and or and in , we find that
We see that scalar triple products are equal when the cyclic order of the three vectors is unchanged, here , , .
We are going to use this property in the next example.
Example 2: Calculating the Scalar Triple Product of the Three Unit Vectors and Its Permutations
We have three scalar triple products with the same three vectors, , , and , but in different orders. Scalar triple products with the same three vectors always have the same absolute value, but the same sign only if the three vectors are in the same cyclic order. Here, the cyclic order in the first scalar triple product is and it is the same order in the other two as well.
Therefore, the total value of the given expression is 3 times the value of one of the terms.
To calculate one of the scalar triple products, we can use the fact that , and so
Alternatively, we can use the determinant method:
Property: Scalar Triple Product of Coplanar Vectors
The scalar triple product is the scalar product of with , that is, the scalar product of with a vector that is perpendicular to the plane defined by and .
It results that if , , and are coplanar (i.e., in the same plane), then since the scalar product of two perpendicular vectors is zero.
Inversely, if , then , , and are coplanar.
Let us use this property to solve the next question.
Example 3: Finding Missing Components of Coplanar Vectors
Find the value of for which the four points , , , and all lie in a single plane.
As three noncollinear points define a plane, if , , and are indeed noncollinear, then we need to find the value of for which is in the plane containing, , and .
Let us first check that , , and are noncollinear. For this, we simply check that the cross product of and is not zero (we can take here any two different vectors formed with the three points):
Let us find now the value of for which is in the plane . As the scalar triple product of three coplanar vectors is zero, we need to find the value of for which, for example, . The components of are . Hence,
if , that is, if .
Therefore, the value of for which the four points , , , and all lie in a single plane is 8.
Another useful property of the scalar triple product comes from its geometrical meaning. As said above, the scalar triple product is the scalar product of with . We know that is a vector perpendicular to the plane defined by and whose magnitude is the area of the parallelogram spanned by and .
Let us consider a parallelepiped spanned by , , and . Its volume is the area of the parallelogram spanned by and multiplied by its height as shown in the figure. The height is, in turn, given by , where is the acute angle between vector and .
As is a vector perpendicular to the plane defined by and , we have
A change in the orientation of (up or down) does not change the absolute value of its scalar product with , as illustrated in the figure with the angle between and (down). As , .
The volume of the parallelepiped is, therefore,
Property: Geometric Meaning of the Scalar Triple Product
The absolute value of the scalar triple product of three vectors is the volume of the parallelepiped spanned by the three vectors:
It is worth noting that three coplanar vectors do not define any parallelepiped, and therefore the scalar triple product is zero.
Let us now use this property to find the volume of a parallelepiped.
Example 4: Finding the Volume of a Parallelepiped
Find the volume of the parallelepiped with the adjacent sides , , and .
The parallelepiped is spanned by , , and . Its volume is given by the absolute value of the scalar triple product of the three vectors. Hence,
The volume of the parallelepiped with the adjacent sides , , and is 9 volume units.
In the last example, we are going to find possible values for a missing vector component given the volume of the parallelepiped spanned by three vectors.
Example 5: Finding Missing Vector Components given the Volume of the Parallelepiped Spanned by Three Vectors
The parallelepiped on vectors , , and has volume 48. What can be?
The volume of the parallelepiped spanned by the vectors , , and is given by
The volume of the parallelepiped is 48 volume units; hence,
This equation is verified if or if ; hence, if or
The parallelepiped on vectors , , and has volume 48 if or .
- The scalar triple product of three vectors , , and is defined as
- A scalar triple product is equivalent to a determinant:
- Scalar triple products are equal if the cyclic order of the three vectors is unchanged:
- The scalar triple product of three coplanar vectors is zero. Inversely, if , then , , and are coplanar.
- The volume of the parallelepiped spanned by vectors , , and is given by