# What is the physical meaning of a product of vectors?

My teacher told me that Vectors are quantities that behave like Displacements. Seen this way, the triangle law of vector addition simply means that to reach point C from point A, going from A to B & then to C is equivalent to going from A to C directly.

But what is the meaning of a product of vectors? I cannot imagine how a product of displacements would look like in reality. Also, how do we know whether we need the scalar (dot) product or a vector (cross) product?

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""Also, how do we know whether we need the scalar (dot) product or a vector (cross) product?"" In case You need it really, You will know. – Georg Nov 17 '11 at 12:48
@Georg Can you give an example? – Green Noob Nov 17 '11 at 12:54
– Georg Nov 17 '11 at 13:00
@Georg Still didn't understand what a product of displacement means :( – Green Noob Nov 17 '11 at 13:04
Force is a vector. Displacement is a vector. Their Dot product is a scalar Energy. I find the best way to understand these concepts is by considering meaningful combinations. For the cross product consider angular velocity and displacement. – ja72 Feb 6 '12 at 20:41

## 2 Answers

You seem to look for geometrical meanings. The cross product gives the area of the parallelogram that is spanned by the two vectors as the length of the resulting vector and the direction perpendicular to both vectors. The scalar product gives you information about the component of one vector into the direction of the other.

As Georg said, you will probably know when you need it. I also found that school is making this stuff more complicated than it needs to be by just letting the students memorize particles of information instead of teaching understanding. If you have to stay with memorizing, a pretty clear way for distinguishing scalar and vector product is the result in respekt of the direction of the vectors: the cross product gives the maximal value, if the vectors have a 90° angle between each other and 0 for 0°, the scalar product.

About the meaning: I would not think in displacements. A force has nothing to with displacements for starters. A vector is a scalar quantity with a direction, or even more general, just a bunch of numbers - generally more than 1 - with a certain operation like the possibility to add two vectors.

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Why can we not think in terms of displacements? Aren't they vectors too? – Green Noob Nov 17 '11 at 13:23

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Displacements can be described with vectors, but not every vector is a displacement, so sooner or later you will run into problems. Example: I lean against a wall in x-direction. The force acting on the wall is F = (10,0,0) N. The wall obviously doe not move. Where is the displacement? – mcandril Nov 17 '11 at 13:45

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The displacement corresponding to a product of vectors doesn't have a natural intuitive geometrical connection to the displacements corresponding to the two original vectors. So it's not a useful intuitive way to visualize it, unlike the case with the sum. Also, I want to disagree with everybody who says not to think in displacements. You should understand that there is more than one way in which vectors appear (so they're not always displacements). However, thinking in terms of displacements is a very good way to get an intuitive feeling for some aspects of vectors (not products, though). – Peter Shor Nov 17 '11 at 14:32
@mcandril I think we can think of vectors as displacements. In your example, the wall doesn't move because the resultant of forces is zero that is, the wall exerts a normal force which cancels out your weight. In terms of displacement, it is the same as walking from A to B(your weight) & walking back following the same path backwards from B to A (normal force) & hence you have no apparent change in position. – Green Noob Nov 19 '11 at 14:52
Note that I said that vectors are quantities that behave like displacements. I never said that vectors are quantities that cause displacement. Please clarify. – Green Noob Nov 19 '11 at 14:54
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It is a bit misleading to think of Vectors as displacements. Vectors are abstract mathematical objects that live in a Vector Space over a Field (say Real number field). A vector is a higher-order animal that is obtained when you pour the Field over a Vector group.

Quick and dirty introduction:

1. Build a set with a collection of objects.
2. Establish a relationship between the objects by means of an operation. (Say multiplication).
3. See if it forms a Group. (We assume yes).
4. Now bring in a Field (Set which forms a Group under two operations) and form a new algebraic structure called a Vector Space over a Field by establishing certain combination rules between elements of the Field and the elements of the group. To make life easier we choose a field that has one operation the same as the Group.

The Field serves to fill the "holes" between elements in the Group by giving you the ability to scale vectors. Vector "Products" are obtained by asking the question "how do we make vectors talk to each other"? Inner products yield elements in the Field (scalars) and wedge products yield another vector that is not in the same sub-space as the two original vectors.

How do you know whether a physical system can be represented by an inner or outer product? Well, the easiest way to check is experimentally. For example how do we know if $\vec{F}=q(\vec{v}\times\vec{B})$ and not $q(\vec{B}\times\vec{v})$ ?? This is by experiment.

*Remember that when we measure something, we do so in the Field because our results are numbers.*This is a critical concept.

There is a lot more to say and I'll edit this when I have the time. Abstract Algebra is a beautiful subject. Hope this helps. :)

Edit #1: The triangle law of addition comes out naturally when you write down the rules that result in the formation of a vector space. All these geometric pictures are misleading because they are presented to students as the absolute concept. You can ask the question "Why is a vector represented by an arrow?". My opinion (I have never seen this discussed anywhere) is that by giving a "direction" you inherently establish an ordering within the set. A lot more can be said if you think deeper, but I guess I have confused the OP already. :) :)

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Thanks for answering but I didn't understand anything because I'm not familiar with abstract algebra :( And further, since we are using vectors in Physics, I believe there must be a physical way of looking at vectors rather than an abstract mathematical one. – Green Noob Nov 17 '11 at 13:29
Just think about the steps I listed and you will get it eventually. I don't know if there is a physical way of looking at vectors without running into problems down the line. :) What I wrote is probably at a much higher level than you are accustomed to, but it never hurts to get your feet wet if you really want to understand what the deal is? – Antillar Maximus Nov 17 '11 at 13:34
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