vectors

[ydoaPs]

a vector is given by [math]\vec{v}=\sum_{j=1}^{n}a_j{\vec{i_j}}[/math] where i is the unit vector for a given dimension and n is the number of dimensions of the space in which the vector exists. the magnitude ([math]|\vec{v}|[/math]) is given by [math]|\vec{v}|^2=\sum_{j=1}^{n}a_j^2[/math]. the unit vector in the direction of the vector v is given by [math]\vec{u}=\frac{\vec{v}}{|\vec{v}|}[/math]. if theta is the angle between vectors A and B, then [math]{\vec{A}}*{\vec{B}}={|\vec{A}|}{|\vec{B}|}{\cos{{\theta}}}=\sum_{j=1}^{n}a_jb_j[/math].

[math]\cos{\theta}=\sum_{j=1}^{n}\frac{a_jb_j}{|\vec{A}||\vec{B}|}[/math]. the projection of A on B is [math]Proj_{\vec{B}}{\vec{A}}=|\vec{A}|\cos{\theta}[/math]

 

the cross product of two vectors A and B(sorry, i don't know how to do matricies in [math]\LaTeX[/math]) is given by a matrix in which the first row is the unit vector for each dimension, the second row is the corresponding coefficient of the unit vector for the first vector, and the last row is the corresponding coefficient of the unit vector for the second vector.

 

what is a projection and how do you know when to use dot product or cross product?

[mezarashi]

A projection is like a shadow. A projection of A onto B would be like the shadow of A on B, which is different from the dot product, because there is no multiplication with the magnitude of the B vector. You are simply outlining how much of the vector A is in the same direction as B. The trickey thing to do is usually finding the angle between A and B.

 

The cross and dot product are mathematical operations that you can perform on vectors. There are many physical interpretations. As such they are useful in helping us find solutions to physical problems which are accurately modeled by them. For example the Lorentz force (a force vector) on a moving charge in a magnetic field is the cross product of the charge's velocity vector and the magnetic field vector.

[ydoaPs]

can you give some graphs.graphs make things much easier.

 

when do you use dot and when do you use cross?

[matt grime]

just think it through:

 

let u be a vector, we want to find the projection of u onto v, that is we want to find vectors a and b such that u=a+b and a is a vector parallel to v and b is a vector perpendicular to v. the projection is then the vector a. we want to kill b so we'd better dot with v since dotting with orthogonal vectors kills them.

[ydoaPs]

so, cross with orthogonal and dot with proportional?

[DQW]

so, cross with orthogonal and dot with proportional?

What does that mean ?

 

You do not use the cross production at all in finding the projection of u onto v. The projection is given by

 

[math]proj (\vec{u} ~on~ \vec{v}) = |\vec{u}| cos(\theta) \hat{v} [/math]

where [imath]\hat{v}[/imath] is the unit vector along [imath]\vec{v}[/imath]. So, it can also be written as :

 

[math]proj (\vec{u} ~on~ \vec{v}) = \frac{(\vec{u} \cdot \vec{v})}{|\vec{v}|} \hat{v} = \frac {(\vec{u} \cdot \vec{v})}{|\vec{v}|} ~\frac{\vec{v}}{|\vec{v}|}[/math]

 

If you've done Newtonian Mechanics, this is exactly what you do when you resolve vectors (forces) along a pair of orthogonal directions. You find the projections of those vectors onto the required pair of orthogonal unit vectors.

[ydoaPs]

What does that mean ?

sorry for the confusion. there is more than one question being asked.

1)what is projection and what does it look like graphically?

2)when do you use cross product and when do you use dot product?

 

now, i'm going to add:

3)how do you write a matrix in [math]\LaTeX[/math]?

[DQW]

1. http://mathworld.wolfram.com/VectorSpaceProjection.html

 

[DQW]

2. The cross-product of a pair of vectors is a vector that is normal to the plane containing the two vectors and whose length is given by |u||v|sin(theta) {which is the area of the parallelogram made by u and v}. If two vectors are parallel, their cross product is a null vector, so it may be useful to use the cross product to check parallelism.

 

The dot product is a scalar whose value is given by |u||v|cos(theta). When two vectors are perpendicular to each other, their dot product os zero, so it is useful to test orthogonality using the dot (or inner) product.

 

You can also use either product to determine the angle between a pair of known vectors, but since the dot product is easier to calculate, one tends to use that.

[DQW]

3. [imath]\LaTeX[/imath] for matrices : see example below

 

[math]

\left(

\begin{array}{cc}

1 & 0\\

0 & -1

\end{array}

\right)

[/math]

 

click on the matrix to see code.

[ydoaPs]

so, what does the scalar mean? i know what a scalar is, i want to know what this one is.

 

test:[math]{\vec{A}}x{\vec{B}}=\left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k}\\a_1 & a_2 &a_3\\b_1 & b_2 & b_3\end{array}\right|[/math]

 

 

edit:nice, it works

[matt grime]

What does what scalar mean? and why did you then write a deteminant that gives a vector? (you do not need to underline, over line or use arrows for vectors, we know they are vectors, you told us they are vectors, hence in this thread i,j,k are vectors and you don't need to embellish them).

[ydoaPs]

What does what scalar mean?

the dot product

 

and why did you then write a deteminant that gives a vector? (you do not need to underline, over line or use arrows for vectors, we know they are vectors, you told us they are vectors, hence in this thread i,j,k are vectors and you don't need to embellish them).

i was just testing it out.

[matt grime]

So, you're asking what does the dot product mean? Well, what does anything "mean" in mathematics? What does "mean" even, well, mean?

 

The dot product is a simple operation on vectors, and it can be used for many things. I've no idea what it "means", and I doubt that that question even makes sense.

 

I know what it means when x.y is zero and both x and y are not the zero vector. I know what it means when x.y is zero for all y. But I've no idea what the dot product means. (I know what it is...)

 

Just remember, x.y is linear in both components, if x and y are perpendicular then x.y=0, and finally that x.x=|x|^2 and that's it, a simple definition.

[ydoaPs]

ok, i'll rephrase it. what do you use a dot product if you are not testing for orthogonality? why would you use multiplying to get a scalar over multiplying to get a vector or vice versa?

[ydoaPs]

so, projection is the part of a vector(like splitting into x and y) in the direction of another?

when doing physics problems, we often use Fcos(theta) to find the force in the x direction. this is a projection of F on the x-axis, right?

[Dave]

ok, i'll rephrase it. what do you use a dot product if you are not testing for orthogonality? why would you use multiplying to get a scalar over multiplying to get a vector or vice versa?

 

So basically, you're asking for the uses of the dot product

 

Its main use is obviously checking for orthogonality. Other than that, I've seen it used in loads of other things. One of the more interesting uses is for things like line integrals. Less interesting uses are to do purely with notation, i.e. in conjuction with the nabla operator:

 

[math]\nabla \cdot \mathbf{v} = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \cdot (v_1, v_2, v_3) = \frac{\partial v_1}{\partial x} + \frac{\partial v_2}{\partial y} + \frac{\partial v_3}{\partial z}[/math]

 

This is called the divergence. From a less practical standpoint, you can find other functions with the same properties as the dot product (called inner products) and build inner product spaces from them. So in all, they're fairly interesting to study purely from a research point of view. But from a vector calculus standpoint, orthogonality is their biggest application.

[ydoaPs]

so, if you just need to multiply vectors use cross product?

[Dave]

It depends what you want to do with them - I'm not sure why you'd want to multiply them anyway tbh. The cross product will simply give you a vector orthogonal to the two you're crossing.

[matt grime]

so, if you just need to multiply vectors use cross product?

 

 

"multiply" makes no sense at all in this context. explain what you mean by multiply.

 

the dot and corss product are just operations nothign more, nothign complicated. they have many uses and you'll get used to them with prcatice. dotting is useful when you want to exploit orthogonality and corssing when you want to exploit parallelism (if that's a word). there are many other uses to that have little to do with this geometry.

[ydoaPs]

so, just use dot if there is a dot and cross if there is an x and don't worry about it?

[CPL.Luke]

I think he's getting confused because there called products.

[matt grime]

use dot if there is a dot and cross if there is an x

 

the answer is either "trivially, yes" or "i've no idea what your asking now"

[ydoaPs]

i just want to know why use dot product? why use cross product?

[matt grime]

because it works, that is the only reason why you ever use anything in maths. sounds crappy i know but it is true. we;ve pointed out their properties and it is up to you to figure out how to use them. there are so many ways to do so that we cannot begin to describe them to your satisfaction.

 

let;s try abn example. Let x be some vector and n a unit vector. We know that we can write

 

x=tn+m

 

where t is a real number and m is a vector orthogonal to n. find t (in terms of n and x and the other things we are discussing). please try this question.