Triple Scalar Product: Definition, Formula & Example

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  • 0:04 A Triple Scalar Product
  • 0:21 Organizing the Vector Products
  • 1:03 The Cross Product
  • 3:44 Some Interesting Facts
  • 5:02 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Vectors and scalars are often discussed in physics. In this lesson, we define a particular multiplication of three vectors called the triple scalar product and use an example to show how it is calculated.

A Triple Scalar Product

What if you see vectors everywhere? Well, maybe not everywhere. But what if a picture of three kittens reminds you of a special three-vector product?


Three kittens.


One such product is called the triple scalar product. In this lesson, we'll explore this unique combination of vectors.

Organizing the Vector Products

Quantities, like mass and volume, are scalars. A scalar has magnitude but no direction. A vector, like force or velocity, has both magnitude and direction. Imagine multiplying three vectors together and getting a scalar. This happens in the triple scalar product.

Those kittens in the photo are organized as two of one kind and one of another. This is similar to the triple scalar product, where we take a cross product of two of the three vectors. This cross product gives us a new vector. Then we take the dot product of this new vector with the remaining vector. The overall result is a scalar. This sounds more complicated than it is. We'll take it step by step. It's certainly easier than herding kittens.

The Cross Product

When we take the cross product of two vectors, ⃗a and ⃗b, we get a new vector. To show this in a general way, let's say the vector ⃗a is written with components ax, ay and az. Similarly, the vector ⃗b is written with components bx, by and bz. The unit vectors i, j and k complete the description, as you can see:


The vectors a and b


A convenient way to calculate the cross product is to build a matrix using the components of the vectors. Then the determinant of the matrix gives us the cross product. Here's how we build the matrix. The first row of the matrix has the unit vectors. The second row contains the components of the vector ⃗a. The components of the vector ⃗b are in the third row. Here's the cross product of ⃗a and ⃗b appearing here:


The cross product using the determinant.


When we expand this determinant, the resulting cross product is this new vector:


The expanded cross product.


Now we take the dot product with the vector ⃗c. In a dot product, the i components of each vector are multiplied together. In our general case, the i component of the ⃗c vector is cx, and the i component of the cross product is (aybz - azby). This gives us the scalar cx(aybz - azby). We do the same thing with the j components and the k component. Adding these three scalar products together gives us a scalar. Now we have the triple scalar product. It looks like the formula appearing on your screen right now:


The expanded cross product.


A fascinating observation can be made. The result we have is the same as the determinant of the matrix whose rows are the components of the vectors ⃗c, ⃗a and ⃗b.

Some numbers will help clarify this last idea. For example, if the vectors are the ones appearing here, we can clarify the result that's the same as the determinant of the matrix whose rows are the components of the vectors we mentioned before.


The expanded cross product.


Can you read off the components of the ⃗c vector? If you said (1,1,4) you're absolutely correct. The parentheses is a convenient way to group the components of a vector. How about the components of the ⃗a vector? Right. The components of the ⃗a vector are (2, 0, 0). Components of ⃗b? (-1, 3,0).

We can compute the triple scalar product using the following:


The expanded cross product.


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