Multiplying vectors

We've focused so far on how to add and subtract vectors. This is straightforward, since vectors are mathematical structures that behave like spatial displacements. Adding means just doing one after the other, just like walking successive displacements. Subtracting means flipping the direction (multiplying by -1) and adding. But can we multiply vectors?

It turns out that we can, and that the different ways that vectors can be multiplied serve to pick out physically essential elements of pairs of vectors that are combining to produce an important physical result. Two examples include:

• The concept of work — when a force moves an object  resulting in a change in its energy of motion (kinetic energy). In this case what matters is the part of the force that point along (or opposite to) the displacement. (Needs the dot product)
• The concept of torque — when a force moves an object causing it to tend to rotate around a center some distance away. In this case what matters is the part of the force that is perpendicular to the displacement to the center of rotation. (Needs the cross product)

We prefer to introduce math in the contexts where needed so we will discuss the conceptual ideas behind the dot and cross products when we discuss work (and the Work-Energy Theorem) and rotational motion. Here, we present the mathematical definitions of the dot and cross products to serve as references for when those topics are introduced in physics contexts.

The mathematics of matrix multiplication

Suppose we have two 3D vectors that we will write as triplets $(a_x, a_y, a_z)$ and $(b_x, b_y, b_z)$, suppressing for now the $\hat{i}, \hat{j}$ and $\hat{k}$ that specify the $x$, $y$, and $z$ directions respectively. Can we multiply these vectors?

Each vector is made up of three numbers, so in principle we could multiply them in 9 different ways. We could create the nine products $a_xb_x, a_xb_y, a_xb_z, a_yb_x,..,a_zb_z$. This is pretty much the mathematical answer — but we are not just interested in "what is it we can do" but in "what has physical meaning". It turns out that combinations of these nine products have important physical relevance.

The dot product

The combination of products of three pairs

$$\overrightarrow{a} \cdot \overrightarrow{b} = a_xb_x + a_yb_y + a_zb_z  $$

turns out to have the nice (if surprising) property that it's actually a scalar: that is,  changing our choice of axes changes each of the numbers that goes into this calculation, but the combination stays the same! (It's even true if we are in 2D and only have the first two terms.) This combination is called the dot product of the two vectors. It comes up when we want to take the component of one vector in the direction of another.

Physically, the dot product is useful when we want to take the part of Newton's second law that tells us about how the object changes its speed. We know that forces in the direction of motion or against it are what change the speed — forces perpendicular to the motion only change the direction. So we want to take the dot product of the vector form of Newton's second law with the velocity (or the displacement). This yields the Work-Energy Theorem, the key to our development of the concept of energy.

The dot product also appears when we want to calculate the flow through a surface that is not perpendicular to it. Only the flow perpendicular to the surface (in the same direction as the normal to the surface) takes fluid through. The flow parallel to the surface (perpendicular to the normal to the surface) just runs the fluid parallel to the surface.

The cross product

Other combinations of our 9 products from our two vectors can be combined to create a vector known as the cross product. The x component of the vector is made up of the y and z components of the two vectors and so on, like this:

$$(\overrightarrow{a} \times \overrightarrow{b})_x = a_yb_z - a_zb_y $$

$$(\overrightarrow{a} \times \overrightarrow{b})_y = a_zb_x - a_zb_x $$

$$(\overrightarrow{a} \times \overrightarrow{b})_z = a_xb_y - a_yb_x $$

These three combinations behave like the components of a vector when we change coordinate systems. (Though they behave differently when you look at them in a mirror.)

The cross product comes up in physics when we want the component of a vector that is perpendicular to another vector, such as when we are looking at forces on extended objects. The component of the force that is along the line from the pivot point tries to stretch or squeeze the object, but doesn't tend to rotate it around the pivot. Only the component perpendicular to the line from the pivot tends to rotate the object. This is the motivation for introducing the concept of torque. The cross product also comes up in the construction of magnetic forces.

What about the rest?

We have found 4 combinations of products of vector components useful, but there we started with 9 objects. Shouldn't there be 5 more combinations? In fact there are. They don't form either a scalar or a vector, but something different — a symmetric traceless matrix. These mathematical structures are useful in making transformations of vectors, but are beyond the scope of this class. Studying them is the step into the mathematics known as tensor analysis. This is useful in understanding how quantum electronic orbitals combine in chemistry and in advanced mechanical engineering. This sort of reasoning also plays an important role in Einstein's special and general theories of relativity.

For more details on the dot and cross products, read the follow-on pages.

Joe Redish 11/6/11