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.)

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Tuesday, September 2nd, 2008 at 5:57 pm and is filed under 275: Calculus III. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

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

I learned of this problem through Su Gao, who heard of it years ago while a post-doc at Caltech. David Gale introduced this game in the 70s, I believe. I am only aware of two references in print: Richard K. Guy. Unsolved problems in combinatorial games. In Games of No Chance, (R. J. Nowakowski ed.) MSRI Publications 29, Cambridge University Press, 1996, pp. […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can always contact one of your friendly edit […]

The problem is in the quantifiers that are implicit in the statement you are making. What you have is that for all $\epsilon>0$ and all integers $k,m$ with $k>m>0$, there is an $N$ such that if $n>N$, then $|a_n|

The relevant search term is ethnomathematics. There are several journals devoted to this topic (for instance, Revista latinoamericana de etnomatemática). Browsing them (if you have access to MathSciNet, the relevant MSC class is 01A70) and looking at their references should help you get started. Another place to look for this is in journals of history of mat […]

Some of the comments in the previous answers make a subtle mistake, and I think it may be worth clarifying some issues. I am assuming the standard sort of set theory in what follows. Cantor's diagonal theorem (mentioned in some of the answers) gives us that for any set $X$, $|X|

For $\lambda$ a scalar, let $[\lambda]$ denote the $1\times 1$ matrix whose sole entry is $\lambda$. Note that for any column vectors $a,b$, we have that $a^\top b=[a\cdot b]$ and $a[\lambda]=\lambda a$. The matrix at hand has the form $A=vw^\top$. For any $u$, we have that $$Au=(vw^\top)u=v(w^\top u)=v[w\cdot u]=(w\cdot u)v.\tag1$$ This means that there are […]

That you can list $K $ does not mean you can list its complement. Perhaps the thing to note to build your intuition is that the program is not listing the elements of $K $ in increasing order. Indeed, maybe program 20 halts on input 20 but only does it after several million steps, while program 19 doesn't halt on input 19 and program 21 halts on input 2 […]

Have we gotten our Homework 2 assigned yet?

Hi William,

HW 2 is posted now under syllabus.

Thanks a bunch