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?
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:
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:
When we expand this determinant, the resulting cross product is this new vector:
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:
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.
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:
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We see that it ultimately equals 24. Do you see how the components of the vectors are placed in the matrix? Do you see how the determinant gives a scalar answer? Do you know where the three kittens have wandered off to?
Some Interesting Facts
If we repeat the pattern of the vectors ⃗c, ⃗a and ⃗b, we'd get ⃗c ⃗a ⃗b ⃗c ⃗a ⃗b and so on. If we start with any of the three vectors while keeping this order, then we're keeping the cyclical order the same. For the triple scalar product, ⃗c(⃗ax ⃗b) is equal to ⃗a(⃗bx ⃗c), which is equal to ⃗b(⃗cx ⃗a).
The triple scalar product is equivalent to multiplying the area of the base times the height. This is the recipe for finding the volume. In fact, the absolute value of the triple scalar product is the volume of the three-dimensional figure defined by the vectors ⃗a, ⃗b and ⃗c. This figure is called a parallelepiped. It's a figure with three sets of equal parallel faces where each face is a parallelogram. The simplest of these figures is a cube where each face is a square. We take the absolute value because the volume is a positive quantity and the cross product could be positive or negative. Using the numerical three vectors from our example, here's a picture of the resulting parallelepiped:
The three vectors define the parallelepiped.
Do you see how the three vectors define a corner of the figure? I wonder what it would take to get three kittens to stay in one corner.
Let's take a couple moments to review the things that we've learned in this lesson. First, we've got to remember that quantities like mass and volume are scalars, and a vector, like force or velocity, has both magnitude and direction. The triple scalar product produces a scalar from three vectors. The dot product of the first vector with the cross product of the second and third vectors will produce the resulting scalar. The determinant of a matrix made from the components of the three vectors is a convenient way to calculate the triple scalar product. If the cyclical order of the three vectors is maintained, the triple scalar product can be expressed in three different ways. The absolute value of the triple scalar product is equal to the volume of the parallelepiped formed by the three vectors.
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