Well, so far all the book has said on this is that ##\vec u \vec v = -\vec v \vec u## due to an argument with Pythagorean's theorem.
I skipped ahead a little bit, but according to the book ##\vec{u} \vec{v} = \vec{u} \cdot \vec{v} + \vec{u} \wedge \vec{v}## which makes sense from an algebraic...
Well, the point of the chapter is "Geometric multiplication of vectors" in which you ask what is ##\vec{v}^2## is, using the definition ##\vec{v} = |\vec{v}|\hat{v}## we see that $$\vec{v}^2 = (|\vec{v}|\hat{v})(|\vec{v}|\hat{v}) = |\vec{v}|^2\hat{v}^2 = |\vec{v}|^2$$ iff we define ##\hat{v}^2 =...
It's a problem from a book that has a section on hyperbolic numbers (https://www.amazon.com/dp/1704596629/?tag=pfamazon01-20), and you're correct, I should've wrote ##\vec{v}^2 = |\vec{v}|^2\hat{v}^2## and define (for euclidean vectors) that ##\hat{v}^2 = 1##.
But thanks for confirming it...
Not sure how to show that because ##\vec{v} = |v|\hat{v} = 3|e|\hat{e}##, but since ##\vec{e}## is a unit vector we know ##|e| = 1## so our equation now becomes ##\hat{v} = \frac{3\hat{e}}{|v|}##. So, we're left to the task of showing that ##|v| = 3## in order to conclude that ##\hat{v} =...
Ah, how foolish of me! Thanks for linking the two ideas for me. Variable substitutions are truly just function compositions. I've never thought about it that way.
I'm currently typing up some notes on topics since I have free time right now, and this question popped into my head.
Given a problem as follows:
Find the first five terms of the Taylor series about some ##x_0## and describe the largest interval containing ##x_0## in which they are analytic...
The way I look at it, I have a beam that goes into an analyzer on the Z axis, then I take the spin down of that beam and send it to the analyzer on the X axis, so I can't forget that ##\frac{9}{34}## probability of being in the spin up on the Z axis. So if I include that in my calculations, I...
Homework Statement
A beam of spin ##\frac{1}{2}## particles is prepared in the state: ##|\psi> = \frac{3}{\sqrt{34}}|+> + \frac{5i}{\sqrt{34}}|->##
a) What are the possible results of a measurement of the spin component ##S_z##, and with what probabilities would they occur?
b) Suppose that the...
Homework Statement
Show that ##[\hat{L} \cdot \vec{a}, \hat{L} \cdot \vec{b}] = i \hbar \hat{L} \cdot (\vec{a} \times \vec{b})##
Homework Equations
##[\hat{L}_i, \hat{L}_j]= i \hbar \epsilon_{ijk} \hat{L}_k ##
The Attempt at a Solution
[/B]
Maybe a naive attempt, but it has been a while. I...
So, I can see my issue was not taking the hodge dual right, and forgetting about my scale factors for my V. I still have an issue. So, this might be a long post...
Let's derive it for an arbitrary orthogonal coordinate system.
##\omega = h_uh_vh_w du \wedge dv \wedge dw## and our...