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Multiplying vectors

Page history last edited by Joe Redish 12 years, 4 months ago

Class content > Kinematics > Vectors  

 

Prerequisites

 

We've focused so far on how to add and subtract vectors. This is straightforward, since vectors are mathematical structures modeled to 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? 

 

Suppose we have two 3D vectors that we will write as triplets (ax, ay, az) and (bx, by, bz), suppressing for now the i-hat, j-hat, and k-hat that specify the x, y, and z directions respectively. Can we multiply these vectors? Well, there are three numbers for each vector, so in principle we could multiply them in 9 different ways. We could create the nine products axbx, axby, axbz, aybx,... This is pretty much the answer -- but since we are not just interested in "what is it we can do" but in "what is it that might have physical meaning", it's actually different combinations of these nine products that turn out to have physical relevance.

 

The dot product

The combination of products of three pairs

 


 

turns out to have the nice (if surprising) property that it's actually a scalar. 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 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:

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 whenever 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 advanced mechanical engineering and 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.

 

Follow-ons

 

Joe Redish 11/6/11

 

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