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Orthogonal Vectors in Rn Problem

  • #1

Homework Statement


Given ##a\neq b## vectors of ##\mathbb{R}^n##. Determine ##c## which lies in the line segment ##[a,b]=\{a+t(b-a) ; t \in [0,1]\}##, such that ##c \perp (b-a)##. Conclude that for all ##x \in [a,b]##, with ##x\neq c## it is true that ##|c|<|x|##.

Homework Equations


The first part of the question can be solved just using inner product definition.
I don't know how the second part can be solved. But I think could be useful the Cauchy-Schawarz inequality ##|<x,y>| \leq |x||y|## or maybe the cosine rule...

The Attempt at a Solution


If we suppose that ##c \in [a,b]## then we can write ##c## as:
$$c=a+t_0 (b-a)$$
where ##t_0 \in [0,1]##.
By the fact that ##c \perp (b-a)## we have:
$$<c,(b-a)> \quad = \quad0$$
$$\Rightarrow \quad <a+t_0 (b-a),(b-a)> \quad = \quad 0$$
$$\Rightarrow \quad <a,(b-a)>+<t_0 (b-a),(b-a)> \quad =\quad 0$$
$$\Rightarrow \quad <a,(b-a)>+t_0 |b-a|^2=0$$
$$\Rightarrow t_0 = - \frac{<a,(b-a)>}{|b-a|^2} $$

Then ##c## can be uniquely determined as ##c=a+t_0 (b-a)##, where ##t_0 = - \frac{<a,(b-a)>}{|b-a|^2}##.
$$\ldots$$
Now we need to prove that if we consider any other ##x \neq c## in the line segment ##[a,b]## then ##|c|<|x|##.

upload_2018-7-12_16-35-57.png
 

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Answers and Replies

  • #2
vela
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You can show that ##x = c + k(b-a)## for some ##k \ne 0##. Then calculate ##\langle x,x \rangle##.
 
  • #3
Ray Vickson
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Homework Statement


Given ##a\neq b## vectors of ##\mathbb{R}^n##. Determine ##c## which lies in the line segment ##[a,b]=\{a+t(b-a) ; t \in [0,1]\}##, such that ##c \perp (b-a)##. Conclude that for all ##x \in [a,b]##, with ##x\neq c## it is true that ##|c|<|x|##.

The premise of the question is wrong: if ##a \neq b, \; a,b \in \mathbb{R}^n## it is not necessarily true that ##c \perp (b-a)## for some ##t \in [0,1].## For example, if ##a = (1,1)## and ##b = (2,1)##, we have ##b-a = (1,0)## and ##c = (1,1)+t(1,0) = (1+t,1)##. In order to have ##c \perp (b-a)## we need ##t = -1##, so the"perpendicular" ##c## is not on the segment from ##a## to ##b##; it is outside that segment, but still on the line through ##a## and ##b##.

However, if you allow values ##t < 0## and ##t > 1##---in other words, if you allow any ##t \in \mathbb{R}##---then the "minimization" result is true. The easiest way to get that is to minimize ##F(t) \equiv \| c \|^2##, which is a quadratic function of ##t##.
 
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  • #4
The premise of the question is wrong: if ##a \neq b, \; a,b \in \mathbb{R}^n## it is not necessarily true that ##c \perp (b-a)## for some ##t \in [0,1].## For example, if ##a = (1,1)## and ##b = (2,1)##, we have ##b-a = (1,0)## and ##c = (1,1)+t(1,0) = (1+t,1)##. In order to have ##c \perp (b-a)## we need ##t = -1##, so the"perpendicular" ##c## is not on the segment from ##a## to ##b##; it is outside that segment, but still on the line through ##a## and ##b##.

However, if you allow values ##t < 0## and ##t > 1##---in other words, if you allow any ##t \in \mathbb{R}##---then the "minimization" result is true. The easiest way to get that is to minimize ##F(t) \equiv \| c \|^2##, which is a quadratic function of ##t##.
Very well noticed Ray Vickson!
Ok I can reformulate the question as follow:

Homework Statement


Given ##a\neq b## vectors of ##\mathbb{R}^n##. Determine ##c## which lies in the line ##r## determined by in the line segment ##[a,b]=\{a+t(b-a) ; t \in [0,1]\}##, such that ##c \perp r##. Conclude that for all ##x \in r##, with ##x\neq c## it is true that ##|c|<|x|##.

The solution I wrote before can be used in the same way, i.e:
$$c = a + t_0 (b-a) \quad where \quad t_0 = - \frac{<a,(b-a)>}{|b-a|^2}$$

But regarding the last part, i.e., to prove that ##|x|<|c|## for all ##x \in r##, I have no clue how can I minimize the function ##F(t) \equiv \| x \|^2 \equiv \| a+t(b-a) \|^2##.
 
  • #5
You can show that ##x = c + k(b-a)## for some ##k \ne 0##. Then calculate ##\langle x,x \rangle##.
In fact vela, we can write any ##x## in the line determined by the line segment ##[a,b]## as ##x = c + k(b-a)##.
But follow your tip I stucked at the following red question mark:

upload_2018-7-12_19-58-37.png
 

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  • #6
vela
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Don’t substitute for ##c##.
 
  • #7
Ray Vickson
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Very well noticed Ray Vickson!
Ok I can reformulate the question as follow:

Homework Statement


Given ##a\neq b## vectors of ##\mathbb{R}^n##. Determine ##c## which lies in the line ##r## determined by in the line segment ##[a,b]=\{a+t(b-a) ; t \in [0,1]\}##, such that ##c \perp r##. Conclude that for all ##x \in r##, with ##x\neq c## it is true that ##|c|<|x|##.

The solution I wrote before can be used in the same way, i.e:
$$c = a + t_0 (b-a) \quad where \quad t_0 = - \frac{<a,(b-a)>}{|b-a|^2}$$

But regarding the last part, i.e., to prove that ##|x|<|c|## for all ##x \in r##, I have no clue how can I minimize the function ##F(t) \equiv \| x \|^2 \equiv \| a+t(b-a) \|^2##.
As I said already, that is a quadratic function of ##t##. Just expand out ##\| a + (b-a)t \|^2##, using the definition of ##\| \cdot \|^2.##
 

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