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I have an old exam question I need help with before my upcoming exam:

Let $u = (x_1, y_1)$ and $v = (x_2,y_2)$.

Define an inner product in $\mathbb{R}^2$, $\langle u, v\rangle =2x_1x_2+x_1y_2+x_2y_1+3y_1y_2$.

Now I want to find the distance between the line $x+y=1$ and $(0,0)$

My attempt: Let $u=(x,y)$ be the vector we're looking for, $\langle u, u\rangle =2x^2+2xy+3y^2$, call this $B$, so the norm is $\frac{1}{\sqrt{B}}$.

I think there should be a way to do this by diagonalizing the matrix $A$, which is the matrix for our polynomial. So we get $A=T^{-1}DT$, then make a variable-substitution $(u,v)=(x,y)T$ to eliminate the $2xy$-term in our polynomial, then use $x+y=1$ to solve for the smallest value. Is this correct?

Is there any shorter solution I can do? Like using $x+y=1$ directly on $2x^2+2xy+3y^2$?

Bare with my english, but plz let me know where my mathematical notation is wrong!

Thx

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In future, you can use Latex commands (in between dollar signs) to make the mathematics look nicer. You can check the edit I just made to see some examples. –  Michael Albanese Jan 6 '13 at 5:20
    
Thank you Michael, I will do that! –  HansDeKling Jan 6 '13 at 5:23
    
That is not an inner product. Is the term $x_1y_1$ correct? –  WimC Jan 6 '13 at 7:50
    
Oh no, you are right. I mistyped it :( Ill edit...Thx! –  HansDeKling Jan 6 '13 at 7:55

2 Answers 2

up vote 1 down vote accepted

The shortest distance between a fixed point $a$ off a line and a variable point $b$ on a line occurs when $b - a$ is perpendicular to the line. In this case the line $x+y=1$ can be expressed with the inner product as $$\ell = \{ (x, y) \mid x+y = 1\} = \{ u \mid \langle u, (2, 1) \rangle = 5 \}.$$ In this last form it is clear that $(2,1)$ is perpendicular to $\ell$. The point closest to $(0,0)$ is therefore the intersection of $\ell$ with the line through $(0,0)$ and $(2,1)$. Then use the inner product again to find the distance between that point and $(0,0)$.

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Thx! I realize now that this wasn't the type of exercise I wanted to ask about. Although, Ive learned sth new! Satisfied. –  HansDeKling Jan 6 '13 at 10:27

I am not sure what you mean by the matrix of a polynomial, so I'm afraid I can't answer the question directly.

A way that makes use of the linear algebra you know is to choose some vector on the line, which will have the form $\mathbf{x}=\begin{bmatrix} x \\ 1-x \end{bmatrix}$. If a vector points along the shortest distance from a point to a line, it will necessarily be orthogonal to the line.

We know that the line has its "slope" vector to be $\mathbf{m}=\begin{bmatrix} 1 \\ -1 \end{bmatrix}$. Therefore, when we set the inner product $\langle\mathbf{x},\mathbf{m}\rangle=0$, we get \begin{align} 2x-x(1-x)+(1-x)-3 &= 0 \\ x^2+2x-2x+1-3 &=0 \\ x^2 &= 2 \\ x &= \pm \sqrt2 \end{align}

Thus, the desired vector is $\begin{bmatrix} \sqrt2 \\ 1-\sqrt2 \end{bmatrix}$, and so the distance is the square root of the inner product with itself: \begin{align} \langle\mathbf{x},\mathbf{x}\rangle &= 2(\sqrt2)^2+2\sqrt2(1-\sqrt2)+3(1-\sqrt2)^2 \\ & = 4 + 2\sqrt2 - 4 + 3(3-2\sqrt2) \\ & = 9 - 4\sqrt2, \end{align}

which is $\sqrt{9-4\sqrt2} = 2\sqrt2 -1$.

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Thx Eric, I didnt think about the fact that $\mathbf{x}=\begin{bmatrix} x \\ 1-x \end{bmatrix}$. This was a much neater way to do it. –  HansDeKling Jan 6 '13 at 6:00
    
What I meant by the matrix of the polynomial was, $(x,y)\begin{bmatrix} 2,1 \\ 1,3 \end{bmatrix}(x,y)^t$. Perhaps I cant say that this is a matrix of a polynomial? –  HansDeKling Jan 6 '13 at 6:04
    
I've never seen the concept used that way before, but I also haven't studied a lot of linear algebra :) –  Eric Stucky Jan 6 '13 at 6:23
    
Aha ok, I had no idea. Never seen a forum in this format before. I appreciate your answer, but Ill wait and see then. The thing is, I believe that some of the questions of this type on my exam will be in such a form that I cant use your solution, so I need some more answers :) Thx again for replying, and telling me about the system of this site. –  HansDeKling Jan 6 '13 at 7:33

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