Solving Cross Product w/ Determinants: Setting Up Equations

[danny271828]

I'm having trouble relating the cross product form |a||b|sin(theta) to its component form (a1b2 - a2b1) ... and so on... I know how to do this mathematically so please don't just suggest some proof that I can find in every textbook... The component form involves the solutions to equations using determinants I believe... I was wondering if anyone could get me going in the right direction as far as setting up a set of equations to solve in order to arrive at this component form... I know I have seen this somewhere but cannot find the right book... So I guess you could say I'm trying to setup the right question, in other words, is there a set of equations for 2 vectors in a plane that can be solved via determinants in order to arrive at this component form for the cross product? I'm having a little trouble stating the question even...

 

[danny271828]

I guess another way of what I am asking is - is there any way to arrive at the component form of the cross product without knowing it is equal to absin(theta)?

 

[radou]

Well, a x b = $$\det \left( \begin{array}{ccc} \textbf{i} & \textbf{j} & \textbf{k} \\ a1 & a2 & a3 \\ b1 & b2 & b3 \end{array} \right)$$, unless you were referring to something else?

(Of course, a = a1i + a2j + a3k, etc.)

 

[danny271828]

well that's kind of what I'm asking... is there a way to arrive at that determinant form without knowing a x b equals |a||b|sin(theta)? So we want to construct a vector that is perpendicular to 2 vectors in a plane and has a magnitude that somehow expresses the amount of rotation.

 

[nicksauce]

"well that's kind of what I'm asking... is there a way to arrive at that determinant form without knowing a x b equals |a||b|sin(theta)?"

Well that's kind of the definition for the cross-product. Nevertheless, maybe this might help a bit:

Start with the following definitions for a right-handed co-ordinate system:

$$\hat{x}\times\hat{x} = \hat{y}\times\hat{y} = \hat{z}\times{z} = 0$$
$$\hat{x}\times\hat{y} = -\hat{y}\times\hat{x} = \hat{z}$$
$$\hat{y}\times\hat{z} = -\hat{z}\times\hat{y} = \hat{x}$$
$$\hat{z}\times\hat{x} = -\hat{x}\times\hat{z} = \hat{y}$$

So if you write

$$A\times B = (A_x\hat{x} + A_y\hat{y} + A_z\hat{z}) \times (B_x\hat{x} + B_y\hat{y} + B_z\hat{z})$$

(Ie: $$A_y\hat{y}\times B_x\hat{x}=A_yB_x(\hat{y}\times\hat{x}) = -A_yB_x\hat{z}$$)

Expand, regroup, and this will lead you to the determinant form.

 

[HallsofIvy]

well that's kind of what I'm asking... is there a way to arrive at that determinant form without knowing a x b equals |a||b|sin(theta)? So we want to construct a vector that is perpendicular to 2 vectors in a plane and has a magnitude that somehow expresses the amount of rotation.

If you don't use "length of a x b= |a||b|sin(theta)" (and the fact that a x b is perpendicular to be a and b with the "right hand rule"- it is not correct that a x b= |a||b|sin(theta)!) the what definition of cross product ARE you using?

Obviously, you have to have some definition before you can derive a formula!

nicksause is using, as a definition, that $\vec{i}\times \vec{j}= \vec{k}$, $\vec{j}\times\vec{k}= \vec{i}$, and $\vec{k}\times\vec{i}= \vec{j}$ together with requiring that the cross product be associative, distribute over vector addition, and be anti-commutative.

 

[nicksauce]

nicksause is using, as a definition, that $\vec{i}\times \vec{j}= \vec{k}$, $\vec{j}\times\vec{k}= \vec{i}$, and $\vec{k}\times\vec{i}= \vec{j}$ together with requiring that the cross product be associative, distribute over vector addition, and be anti-commutative.

Right, I should have mentioned that.

 

[D H]

nicksause is using, as a definition, that $\vec{i}\times \vec{j}= \vec{k}$, $\vec{j}\times\vec{k}= \vec{i}$, and $\vec{k}\times\vec{i}= \vec{j}$

That is the original quaternion-based definition of the cross product. Even the use of $\vec{i}, \vec{j}, \vec{k}$ as unit vectors comes straight from the quaternions. The determinant form is an easy mnemonic for some; I prefer the even/odd permutations of i,j,k (or whatever).

As an aside, the concept of vectors and vector spaces is a relatively recent invention (end of the 19th century). We are introduced to vectors in the first week of freshman physics and use vector-based calculations throughout. How did physicists do things, even very basic freshman-level physics things, before the invention of vectors and all that is associated with them?

 

[robphy]

They probably used systems of equations expressed in some choice of coordinate system. I would imagine that there was more use of geometric and trigonometric arguments.