Cross product, dot product concepts

[teodorakis]

Hi,
I'm very ashamed about getting a fully understanding of these vector product concepts. But i did a lot of search and get an idea about them. I read Feynman's lecture on physics and almost completely understand the mathematics behind their properties. In that book feynman proves that cross product operation produces some new numbers that behaves just like vectors and scalar product produces a number that is independent of coordinate system of our choice, before i read this book i thought that, we define that cross product produces vectors and dot product produces scalars, but as i understand we only define some relation about components of vectors and according to this operations the results came out as vectors or scalars. Everything is fine, but in historical view, when i put myself to those scientists who found this vector analysis:), this idea of finding cross or dot product seems very unnatural and difficult to me, of course maybe i have 10 percent of their iq:), but i don't know I'm just confused, and i don't get vector or cross product like i get f=ma. May be you can advise me some books that explains these concept not mathemtically but intuitively.

 

[teodorakis]

and i know most students just memorise the definiton of torque for example, it's like someone put a gun on your head and say that's the way to understand it, just memorise.In this case this torque definiton is given, fine but can't they explain it like , we got f=ma, emprically, which is a vector equaiton in words, when you aplly a directional quantitiy F to a mass m you will get a change in speed B]in that direction[[/B], so think of a particle in space which has a position r, what do you need to "turn" this particle? and student will ask what is turn? you will say changing the position vector of particle not in magnitude but as position according to some origin, to do this what can you do? You can not put a force and r direction because f=ma says if you do this particles r vector increas in magnitude not changing any position, acoording to f=ma, you have to apply a force that has no component in r direction to make r turn.That is of course perpendicular direction to r.Now we explore why we take the poerpendicular componnet of force in torque.
Then we want to know why we multiply this perpendicular product with position vestor r? At this point i have an explanation but I'm not sure correct, here i go, rotational analog of force is torque we say and by our definition we have to tell this mathematically in our equation, so perpendicularness explained above, now what is this multiplication means it means "to some point":) let me explain for example for simplicity in plane imagine two particles in r and 3r distance to an origin we apply same force with same angle so what happened at the end as we expect one particle begin to sweep 3r radius area and other begin to sweep r radius area, so the vertical force component of 3r causes a 3r area swept and r causes and r area swept, so i try to explain the definiion of torque by "real words" to make sense to me at least i may have some mistakes. may be i just make no sense, but i need an explanation about cross and vector prodcuts like this way.
thank you

 

[mikelepore]

You have a dot product when the answer to a problem becomes maximum to the extent that two things are in the same direction (parallel). For example, if you pull a wagon along the ground with a handle -- assume the wagon is constrained to move horizontally and will not rise into the air -- you will perform the maximum work if the force is applied horizontally. You would be wasting some force if the handle were pulled diagonally upward. Therefore the work done is the dot product between force and displacement. When angle theta is zero, cos theta has the maximum value of 1.

You have a cross product when a magnitude is maximum to the extent that two things are perpendicular, not parallel. When angle theta is a right angle, sin theta has the maximum value of 1.

But the cross product also provides the feature that a resulting vector will be perpendicular to both of the given vectors, in the third dimension.

 

[Repainted]

Well basically, the cross and dot product were created to be useful tools of mathematics. Doing the 'multiplication' of the vectors is only a 'shortcut' to finding the data about the vectors which you need.

For example we have 2 vectors a = a₁i + a₂j + a₃k, and b = b₁i + b₂j + b₃k.

a.b = a₁b₁ + a₂b₂ + a₃b₃, and this is also equal to |a||b|cos$$\theta$$, which you seem to understand since you've read how it works.

The 'multiplication' of the components of vectors like this are just a method of finding |a||b|cos$$\theta$$ without having to actually measure the angle between th vectors, followed by multiplying them by the respective magnitudes.

If you're shocked at how they managed to come up with such elegant mathematics, you're not alone. I remember being really surprised at the keen insight which led to x + iy being expressed in the form re^i$$\theta$$.

Unless what you're looking for is a proof that a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos$$\theta$$ and a similar proof for the cross product?

 

[teodorakis]

in conclusion we make definitions for usefulness, and they're the things we create, they are not fundemantel rules, in force case f=ma is always working but to ease things(i mean complicated motion) we define suc a thing like torque, which tells us the rotaitional character of force mathematically. Another example is curl of a vector field we define it for the change of the vector field in perpendicular direction(very loose definition but something like that), am i get it right?

 

[mikelepore]

in conclusion we make definitions for usefulness, and they're the things we create, they are not fundemantel rules,

The symbolic operations are made up, but they wouldn't describe nature correctly if they were made up differently. As they have been defined, we can express what nature does.

For example, a particle with electric charge q, moving with velocity v through a region of space where there is a magnetic field B, will experience a magnetic force q(vXB). We need to express a couple things: the sin of the angle between v and B, and also the concept of pointing in the third dimension according to the right hand rule. Luckily we already have a symbol that expresses both of those things -- it is the cross product -- so we select the cross product as our vocabulary. If we didn't already have something like the cross product we would have to invent it now.

The remainer of your post, I didn't understand enough of it to be able to have a comment.

 

[teodorakis]

cross prduct concept comes meaningfull when i first think it as a dirceted area, then in three dimensional space we can associate a plane with only one unique axis, which enables us define the cross product as a vectoral quantity. YOu can prove the vectoralness of cross pğrodcut by using it's componentwise definition and then rotate your coordinate system around z axi only (for simplicity) and you will end up with new torques which is related to old torques withvectoral transformation equations(feynman's lecture in physics), so we can prove what we define by cross product is a vectoral quantitiy, then for three dimensional space(cross prodcut isn't defined in 4d space), we can associate one unique axis qith a plane, so you can guess the rest of the story.

 

[Kricket]

I've studied a good bit of mathematics, so I can give you the mathematician's perspective on 1.5 of your questions.

Mathematically, the "dot" product (or inner product) is necessary to introduce the concept of an angle in a metric space. Without it, you only have distance, but no concept of "what's the angle between two vectors?" Just think of it as a neat trick you can use to do stuff like project vectors onto each other.

Consider the integral for Work (F.dr) and the cosine definition of the dot product (A.B = |A||B|cos(theta)). The (|F|cos(theta)) part is equivalent to the projection of F along the r line - in other words, how much of F went that way. The |dr| part is like saying, ok, we've got this force, but I'm not moving a whole unit...only (dr) units...thus I have to multiply by the fraction (dr/1) to scale it down. Add up all those tiny pieces, take the limit, and you've got your integral.

I haven't studied the generalization of the cross product, but I know that it represents the area of the parallelogram formed by the two vectors. The |A||B|sin(theta) part is exactly the formula for the area of a parallelogram. The direction tells you what the axis of spin is. The |A||B| part captures that idea that a force of 2N at a radius of 1m produces the same effect as a force of 1N at a radius of 2m.

Think about torque. When a force F at radius r is applied, you can break that force down into the component orthogonal to r, and the component along r. Only the orthogonal component induces a twisting/rotation of the object - the component along r just acts the same as if something on the outside applied that force linearly. This is what's captured by the (sin(theta)) part of the formula - you're actually throwing away the force along r in this operation.

 

[teodorakis]

I've studied a good bit of mathematics, so I can give you the mathematician's perspective on 1.5 of your questions.

Mathematically, the "dot" product (or inner product) is necessary to introduce the concept of an angle in a metric space. Without it, you only have distance, but no concept of "what's the angle between two vectors?" Just think of it as a neat trick you can use to do stuff like project vectors onto each other.

Consider the integral for Work (F.dr) and the cosine definition of the dot product (A.B = |A||B|cos(theta)). The (|F|cos(theta)) part is equivalent to the projection of F along the r line - in other words, how much of F went that way. The |dr| part is like saying, ok, we've got this force, but I'm not moving a whole unit...only (dr) units...thus I have to multiply by the fraction (dr/1) to scale it down. Add up all those tiny pieces, take the limit, and you've got your integral.

I haven't studied the generalization of the cross product, but I know that it represents the area of the parallelogram formed by the two vectors. The |A||B|sin(theta) part is exactly the formula for the area of a parallelogram. The direction tells you what the axis of spin is. The |A||B| part captures that idea that a force of 2N at a radius of 1m produces the same effect as a force of 1N at a radius of 2m.

Think about torque. When a force F at radius r is applied, you can break that force down into the component orthogonal to r, and the component along r. Only the orthogonal component induces a twisting/rotation of the object - the component along r just acts the same as if something on the outside applied that force linearly. This is what's captured by the (sin(theta)) part of the formula - you're actually throwing away the force along r in this operation.

dear kricket,
Thank you for your kind reply, but you're talking about nothing new about cross product, and torque in torque there's a spin but cross product in general does not define a spin twist or ratoation, spin in torque comes from the relation f=ma, in genreal cross product defines a directional quantitiy related to two vectors, a definition of one vector's perpendicular component over another vector and this is scuh a magnitude that we can associate a unique direction in 3d space. That's the way ı undertand these concepts.

 

[D H]

a = a₁i + a₂j + a₃k

If you're shocked at how they managed to come up with such elegant mathematics, you're not alone. I remember being really surprised at the keen insight which led to x + iy being expressed in the form re^i$$\theta$$.

Notice the similarity here. A complex number is written in the form $x+iy$. A 3-vector, $x\mathbf i + y\mathbf j + z\mathbf k$. Add in a scalar and you have a quaternion: $w+ix + jy+kz$. The reason we use i,j,k as unit vectors is because the way physicists represent vectors is an offshoot of the quaternions. Think of the quaternions as an extension of the complex numbers, where each of i, j, and k are distinct roots of -1. Hamilton was taking a walk when he came up with the idea of an extension of the complex numbers. Afraid he would forget this insight, he carved the fundamental equation into a bridge:

$$i^2=j^2=k^2=ijk=-1$$

Heaviside and Gibbs later developed vector analysis from Hamilton's quaternions. The definition of the dot product and cross product are a direct consequence of quaternion multiplication. Heaviside, Gibbs, and others did not like the quaternions; they liked their vector analysis much better. The vectorialists won the great quaternion war, which raged among mathematicians and physicists for the last third of the 19th century.

 

[Kricket]

dear kricket,
Thank you for your kind reply, but you're talking about nothing new about cross product, and torque in torque there's a spin but cross product in general does not define a spin twist or ratoation, spin in torque comes from the relation f=ma, in genreal cross product defines a directional quantitiy related to two vectors, a definition of one vector's perpendicular component over another vector and this is scuh a magnitude that we can associate a unique direction in 3d space. That's the way ı undertand these concepts.

I was just using torque as an example to illustrate the meaning of the cross product. Spin (or rather, angular acceleration) does indeed come from force; however, in 3 dimensions you have not only how fast you're spinning, but also what you're spinning around (the axis of rotation). The torque vector resulting from a force applied at a given radius, at a given point on the circle, captures everything needed to describe spin: the speed (= magnitude of torque), the axis of spin (= the line on which the torque vector lies), and the direction (i.e. which way is "plus theta" - uniquely defined as long as you stick to either the left- or right-hand rule).

Perhaps an intuitive way to describe it would be to think of a screw - the torque vector goes along the screw, positive rotation makes the screw move one way, negative rotation causes it to move the other way, and as the torque is applied, a sideways force is generated on the threads that causes them to dig into the wood.