In class we tried to find the distance from to the plane of equation .

There are several ways of doing this. For example:

Fix a point on the plane . Any point will do, say .

Find a vector perpendicular to the . For example, .

[Again, if a plane has equation then is perpendiculart to it.

For example, is parallel to the plane , which we can rewrite as , which is the equation of the set of points perpendicular to the vector . This is to say, the plane is perpendicular to the vector . Since the plane : is parallel to , is also perpendicular to .

Another way of reaching the same conclusion is to rewrite in the form for some appropriate vector . There are many choices of and they all work; all we need is that , i.e., that is in the original plane. For example, the point belongs to the plane : , so we can rewrite the equation of as , which is equivalent to saying that But this means that the vector is perpendicular to the vector , which is an arbitrary vector in the direction of the plane.]

Let’s continue with the problem of finding the distance from to :

Consider the projection of the vector in the direction of . Clearly, the distance from to is the length of .

Recall that .

Then and the distance is .

In detail, so and , so .

Notice that (as discussed in class) the distance from a point to the line in the direction of that goes through a point is given by , while (by the above) the distance from a point to a plane containing a point and perpendicular to a vector is given by . While the expressions are similar, one involves a cross product and the other a dot product. This is because in one case we express the distance in terms of the sine of an angle , and in the other, in terms of its cosine or, what is the same, in terms of the sine of . (Drawing a diagram may help you clarify the situation.)

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3 Responses to 275- Distance from a point to a plane

(As I pointed out in a comment) yes, partial Woodinness is common in arguments in inner model theory. Accordingly, you obtain determinacy results addressing specific pointclasses (typically, well beyond projective). To illustrate this, let me "randomly" highlight two examples: See here for $\Sigma^1_2$-Woodin cardinals and, more generally, the noti […]

I am not sure which statement you heard as the "Ultimate $L$ axiom," but I will assume it is the following version: There is a proper class of Woodin cardinals, and for all sentences $\varphi$ that hold in $V$, there is a universally Baire set $A\subseteq{\mathbb R}$ such that, letting $\theta=\Theta^{L(A,{\mathbb R})}$, we have that $HOD^{L(A,{\ma […]

A Wadge initial segment (of $\mathcal P(\mathbb R)$) is a subset $\Gamma$ of $\mathcal P(\mathbb R)$ such that whenever $A\in\Gamma$ and $B\le_W A$, where $\le_W$ denotes Wadge reducibility, then $B\in\Gamma$. Note that if $\Gamma\subseteq\mathcal P(\mathbb R)$ and $L(\Gamma,\mathbb R)\models \Gamma=\mathcal P(\mathbb R)$, then $\Gamma$ is a Wadge initial se […]

Craig: For a while, there was some research on improving bounds on the number of variables or degree of unsolvable Diophantine equations. Unfortunately, I never got around to cataloging the known results in any systematic way, so all I can offer is some pointers to relevant references, but I am not sure of what the current records are. Perhaps the first pape […]

Yes. Consider, for instance, Conway's base 13 function $c$, or any function that is everywhere discontinuous and has range $\mathbb R$ in every interval. Pick continuous bijections $f_n:\mathbb R\to(-1/n,1/n)$ for $n\in\mathbb N^+$. Pick a strictly decreasing sequence $(x_n)_{n\ge1}$ converging to $0$. Define $f$ by setting $f(x)=0$ if $x=0$ or $\pm x_n […]

All proofs of the Bernstein-Cantor-Schroeder theorem that I know either directly or with very little work produce an explicit bijection from any given pair of injections. There is an obvious injection from $[0,1]$ to $C[0,1]$ mapping each $t$ to the function constantly equal to $t$, so the question reduces to finding an explicit injection from $C[0,1]$ to $[ […]

One way we formalize this "limitation" idea is via interpretative power. John Steel describes this approach carefully in several places, so you may want to read what he says, in particular at Solomon Feferman, Harvey M. Friedman, Penelope Maddy, and John R. Steel. Does mathematics need new axioms?, The Bulletin of Symbolic Logic, 6 (4), (2000), 401 […]

"There are" examples of discontinuous homomorphisms between Banach algebras. However, the quotes are there because the question is independent of the usual axioms of set theory. I quote from the introduction to W. Hugh Woodin, "A discontinuous homomorphism from $C(X)$ without CH", J. London Math. Soc. (2) 48 (1993), no. 2, 299-315, MR1231 […]

This is Hausdorff's formula. Recall that $\tau^\lambda$ is the cardinality of the set ${}^\lambda\tau$ of functions $f\!:\lambda\to\tau$, and that $\kappa^+$ is regular for all $\kappa$. Now, there are two possibilities: If $\alpha\ge\tau$, then $2^\alpha\le\tau^\alpha\le(2^\alpha)^\alpha=2^\alpha$, so $\tau^\alpha=2^\alpha$. In particular, if $\alpha\g […]

Fix a model $M$ of a theory for which it makes sense to talk about $\omega$ ($M$ does not need to be a model of set theory, it could even be simply an ordered set with a minimum in which every element has an immediate successor and every element other than the minimum has an immediate predecessor; in this case we could identify $\omega^M$ with $M$ itself). W […]

Have we gotten our Homework 2 assigned yet?

Hi William,

HW 2 is posted now under syllabus.

Thanks a bunch