Confused about the True Geometric Meaning of a Dot Product Answer.

[The Head]

I have performed numerous calculations of dot products throughout my math courses, but I think I lack a fundamental understanding of what it actually means, beyond the abstract way I have been taught to deal with them. I know the definitions (it's the inner product, or the projection of A on to B), but the answer you get with a dot product, does it have a geometric representation? What specifically does it mean?

For example, if you have two vectors (1,4) and (5,0), the dot product is 5. But what does that 5 really mean and can you represent it geometrically. I know it is a scalar and not a vector, but I am hoping there is a way to represent it. Obviously, if you had (2,4) and (5,0) instead, the dot would be twice as large because the angle between them is less and the vectors "share" more in common (a larger first component).

So what I am asking is if there is any true meaning to these dot product numbers of 5 and 10 and is there a way to geometrically represent them? Or, are they merely numbers for calculating the angle in between vectors, where a larger dot product means a smaller angle.

 

[Stephen Tashi]

I know the definitions (it's the inner product, or the projection of A on to B).

Maybe you don't know the definitions. The dot product of A with B has something to do with the projection of A onto B, but it isn't identical to the projection of A onto B. When you read a definition, you must have some sympathy with its author. Even though he is a misguided impractical intellectual, you must try understand exactly what he is saying. Resist the temptation to rephrase definitions "in your own words". That's a good technique to use in many fields of the liberal arts, but not in mathematics.

There are several ways to think about the dot product intuitively. How do you compute the work done by force A on distance B when both are vectors? You multiply the length of the component of A that points in the direction of B times the length of B. Can you see that the calculations that are involved in computing the dot product do that? Is your basic question why the calculation (Ax,Ay) dot (Bx,By) = Ax Bx + Ay By accomplishes the same thing as the calculation (length of A) (cos of angle between B and A) (length of B)?

 

[The Head]

Okay, so thinking about your response leads me to another question. Let's take the example you provided (W=Fd) and the dot product I mentioned. Let's say that F= (1,4) and d= (5,0). Am I correct in saying that this dot product would imply a force of 1N acting for 5m, and another of 4N acting for 0m, leading to a total work of 5J?

What confuses me then is that we represent F as a vector on a two dimensional plane, as it has two components. Then how do we represent two distinct forces, each acting in two dimensions. For example, one force acts by pushing with a magnitude of 1N to the left and 2N in the forward direction for a distance d1, while a second force acts with a magnitude of of 2N to the right and 1N in the forward direction for a distance d2? How would we write these vectors to set up the dot product?

Thank you!

 

[Stephen Tashi]

The dot product of a force F = (1,4) with a distance vector d = (5,0) computes the work work done by force of 1 was acting for a distance of 5 plus a force of 4 acting for a distance of 0.

What confuses me then is that we represent F as a vector on a two dimensional plane, as it has two components. Then how do we represent two distinct forces, each acting in two dimensions. For example, one force acts by pushing with a magnitude of 1N to the left and 2N in the forward direction for a distance d1, while a second force acts with a magnitude of of 2N to the right and 1N in the forward direction for a distance d2? How would we write these vectors to set up the dot product?

I can't visualize that exactly because I don't understand the distinction between "forward" and "right" etc.

There are situations where it is useful to compute the work done by different forces on different distance vectors and add the results together to get "total work". But whether it is useful to do that or not depends on the particular context. Do you mean for these forces to be acting on the same physical object? Are they applied at the same point on that object or at different points?

Questions about forces and work are partly questions about phyics (i.e. reality) not about mathematics. It is physics that tells that that when have a force in certain direction that we are permitted to imagine that force as being produced by two other component forces that are the sides of a parallelogram (or rectangle) that have the original force as its diagonal. It's physics that tells us that if two forces act at the same point on a body then we can add two forces together by the parallelogram law and imagine them to be the single resultant force. There is no mathematical proof that nature must act this way.


If we accept the parallelogram law then it is a question for mathematics whether various methods of calculating resultant forces and component forces give answers that are consistent with the parallelogram law.

So the following type of question is can be answered by mathematics: If I add two vectors together by the parallelogram law using the geometric procedure of representing them as line segments, the constructing the parallelogram and drawing the diagonal, do I get the same answer as I would if I introduced an xy Cartesian coordinate system in the drawing, found the x and y components of the forces and added the respective x and y components together?

Likewise, whether the formula Ax Bx + Ay By computes the same numerical result as the geometric approach to analysing the work done by a force A acting for a distance vector B is a question for mathematics.

Those questions can be answered, essentially by using trigonometry to prove the results of the different methods are same.

 

[lavinia]

I have performed numerous calculations of dot products throughout my math courses, but I think I lack a fundamental understanding of what it actually means, beyond the abstract way I have been taught to deal with them. I know the definitions (it's the inner product, or the projection of A on to B), but the answer you get with a dot product, does it have a geometric representation? What specifically does it mean?

For example, if you have two vectors (1,4) and (5,0), the dot product is 5. But what does that 5 really mean and can you represent it geometrically. I know it is a scalar and not a vector, but I am hoping there is a way to represent it. Obviously, if you had (2,4) and (5,0) instead, the dot would be twice as large because the angle between them is less and the vectors "share" more in common (a larger first component).

So what I am asking is if there is any true meaning to these dot product numbers of 5 and 10 and is there a way to geometrically represent them? Or, are they merely numbers for calculating the angle in between vectors, where a larger dot product means a smaller angle.

The dot product comes from a relationship in Euclidean geometry.

If AB and AC are two line segments that form an angle, let Ax be the orthogonal projection of AB onto the line through AC and Ay the orthogonal projection of AC onto the line through AB.

Similar triangles says that Ax.AC = Ay.AB

These products are the inner product of AB and AC.

The algebraic formula is just Ax.AC or Ay.AB expressed in Cartesian coordinates.

The inner product is a test for orthogonality since if AB and AC are perpendicular then their orthogonal projections, Ax and Ay, are both zero.

More generally Ax/AB or Ay/AC are the cosine of the angle between the two line segments.

In terms of inner products, the cosine is (Ax.AC)/AC.AB or (Ay.AB)/AC.AB. This is the usual formula for the cosine.

 

[homeomorphic]

v dot w is the length of the orthogonal projection of v onto w, times the length of w. That's how I tend to think of it. So, if w is a unit vector, it's just the length of the projection.

It can be a good idea to express definitions in your own words. You just have to be careful not to distort the definition.

Actually, the dot product was discovered by means of the quaternions, so it appeared ready made in essentially its present form. It so happened it was found to have this nice geometric interpretation as projection, times length of the vector you are projecting to.

I was always somewhat dissatisfied with the dot product until I found a good explanation that the definition, it terms of coordinates, coincided with the geometric one. And by explanation, I don't mean ugly calculation, as there were plenty of those available for that purpose.