Jump to content
Science Forums

Vectors


Aki

Recommended Posts

there are 2 standard ways to multiply vectors.

1) The inner product (or dotproduct, indicated by a .):

An inner product can be defined in many other ways, but the standard way (the dot product) is: A.B = a1*b1+a2*b2+ etc (a1 is the first element of vector A, etc). so the result is a scalar.

 

2) the outer product (or cross product, indicated by x)

the cross product is only defined for three dimensions, there is a general version, the so called wedge-product.

the cross product looks like this: C=AxB with: c1=a2*b3-a3*b2. c2=a3*b1-a1*b3, c3= a1*b2 - a2*b1.

so this product gives a vector back.

 

both products have lots of interesting properties of which some are: (|A| = length(A))

- A.B = |A||B| cos (A,:)

- |AxB| = |A||B| sin (A,B) = area of the parallellogram spanned by A and B

- sqrt(A.A)=|A|

 

Bo

Link to comment
Share on other sites

Originally posted by: Tim_Lou

why would you multiply vectors? how is it meaningful?

 

i cant think of a example of "multiplying direction"... well, maybe in complex numbers. :

 

It's just for my curiosity. I don't know where you would use it either

Link to comment
Share on other sites

Bo,

do you know a situation where people need to multiply vectors?

 

about a million

 

you need to understand that there are many physical properties, that are actually a vector (position, velocity, force, (angular) momentum...). Most of the physical formulas that you use on high school would just tell you to take the length of such a vector and use that scalar for the multiplications. however that only works in specific cases. (if the actual product was an innerproduct, it works if the vectors are parallel, if the actual product is an outerproduct, it works if the vectors are perpendicular)

The first example that comes to my mind is the lorentz force: (i use bold for vectors)

a particle with charge q travels with speed v through a magnetic field B. if v and B are perpendicular to each other, there is a force generated, (with direction given by the 'right hand rule' or 'corckscrew-rule') equal to: F=QvB. but notice that actually we are multiplying vectors. and it turns out that the correct formula (that also works if v and B aren't perpendicular) is:

F=Q v x B.

 

other example: Work W is given by the integral |F. ds. Here F is the force, and ds is the path. Here the inner product has to be used.

 

there are many more examples; basicly everything that uses a vector is only correct if you have a sensible definition of a vector product.

 

Bo

Link to comment
Share on other sites

Originally posted by: Bo

The first example that comes to my mind is the lorentz force: (i use bold for vectors)

a particle with charge q travels with speed v through a magnetic field B. if v and B are perpendicular to each other, there is a force generated, (with direction given by the 'right hand rule' or 'corckscrew-rule') equal to: F=QvB. but notice that actually we are multiplying vectors. and it turns out that the correct formula (that also works if v and B aren't perpendicular) is:

F=Q v x B.

Are we actually MULTIPLYING the vectors?

 

As you say, a vector is a scalar amplitude and an angle/ direction. (I am most used to it's application in electrical applications, phase shifts and such) As such when two elements interact, the resultant is scalar wich might be a multiple, but the angle/direction is not multiplied. It's "speed" may multiply, but it's direction is a resultant.

 

Or is this just because of my having dealt with vectors in a 360 degree world? :-)

Link to comment
Share on other sites

Originally posted by: Bo

Bo,

 

The first example that comes to my mind is the lorentz force: (i use bold for vectors)

a particle with charge q travels with speed v through a magnetic field B. if v and B are perpendicular to each other, there is a force generated, (with direction given by the 'right hand rule' or 'corckscrew-rule') equal to: F=QvB. but notice that actually we are multiplying vectors. and it turns out that the correct formula (that also works if v and B aren't perpendicular) is:

F=Q v x B.

 

 

wow, we're learning this in physics class and I didn't realised that we're actually multiplying vectors. lol but this year, we're only dealing with forces perpendicular to each other

 

thanks Bo

Link to comment
Share on other sites

Originally posted by: Freethinker

Originally posted by: Bo

The first example that comes to my mind is the lorentz force: (i use bold for vectors)

a particle with charge q travels with speed v through a magnetic field B. if v and B are perpendicular to each other, there is a force generated, (with direction given by the 'right hand rule' or 'corckscrew-rule') equal to: F=QvB. but notice that actually we are multiplying vectors. and it turns out that the correct formula (that also works if v and B aren't perpendicular) is:

F=Q v x B.

Are we actually MULTIPLYING the vectors?

 

As you say, a vector is a scalar amplitude and an angle/ direction. (I am most used to it's application in electrical applications, phase shifts and such) As such when two elements interact, the resultant is scalar wich might be a multiple, but the angle/direction is not multiplied. It's "speed" may multiply, but it's direction is a resultant.

 

When you multiply two vectors with the inner product you get a scalar which is the amplitude of the projection (the orthogonal projection to be precise) of one vector onto the other. So the angles and the amplitude do interact. This can be shown by the following definition

a.b=|a|*|b|*cos(phi) where phi is the angle between the two vectors.

 

So a and b interacting with a inner product (by the way also called scalar product, for obvious reasons) give a result which depends on both the amplitude and the angle/direction (because phi depends on the angle/direction).

 

I hope it was clear

Link to comment
Share on other sites

Guest
This topic is now closed to further replies.
×
×
  • Create New...