Possible Magnitudes of Vectors in Orthogonal Configurations

[themadhatter1]

Homework Statement

A.
Given unit vectors a, b, c in the x, y-plane such that a · b = b · c = 0,
let v = a + b + c; what are the possible values of |v|?
B.
Repeat, except a, b, and c are unit vectors in 3-space

Homework Equations

The Attempt at a Solution

I have solutions for both that I'm reasonably sure of, I'd just like a 2nd opinion to make sure I solved the problem correctly.

For part A I got |v| = $\sqrt{2}$, $\sqrt{5}$
I took c = <0,1>, a = <1,0> b = <0,1> in this combination |v| = $\sqrt{2}$.
The only other special case is when a = b in this case, |v| = $\sqrt{5}$For Part B I took c = <0,0,1>, a = <1,0,0> and b = $<cos\theta,sin\theta,0>$

Therefore, |v| = $\sqrt{2cos\theta + 3}$, $0\leq\theta\leq 2\pi$

 

[LCKurtz]

The problem with your argument is that you have picked specific, very handy, vectors a,b, and c that satisfy your properties. So you have answered the question for only those three vectors. You have to argue it for any three such vectors.

 

[themadhatter1]

Yes, but since the vectors have to be unit vectors and both vectors a and c are orthogonal to b, |v| is going to be the same regardless. You can rotate a vector by a specific angle and the others have to stay orthogonal so they will move as well by the angle maintaining |v|. That's why I chose the most convenient vectors to work with.

 

[LCKurtz]

Yes, but since the vectors have to be unit vectors and both vectors a and c are orthogonal to b, |v| is going to be the same regardless. You can rotate a vector by a specific angle and the others have to stay orthogonal so they will move as well by the angle maintaining |v|. That's why I chose the most convenient vectors to work with.

Everything you say is true. If this is homework, I would at the minimum include this additional argument. Still, it is just as easy and more elegant to do it with arbitrary vectors in the first place.

 

[quietrain]

i don't understand,

if b.c = 0, then how come b = c? i.e how can b=<0,1> = c ?

 

[quietrain]

why issn't part 1

v = a + b + c
|v| = $\sqrt{a^2 + b^2 + c^2}$ = $\sqrt{3}$ ?

 

[ehild]

Given unit vectors a, b, c in the x, y-plane such that a · b = b · c = 0,


I took c = <0,1>, a = <1,0> b = <0,1>

So you took c=b. Being unit vectors, their scalar product is 1 instead of zero. Which two vectors can be identical instead?

ehild

 

[ehild]

why issn't part 1

v = a + b + c
|v| = $\sqrt{a^2 + b^2 + c^2}$ = $\sqrt{3}$ ?

You do not have 3 independent, mutually orthogonal vectors in 2D. One of the vectors a, b, c is linear combination of the other two.

ehild

 

[vela]

Yes, but since the vectors have to be unit vectors and both vectors a and c are orthogonal to b, |v| is going to be the same regardless. You can rotate a vector by a specific angle and the others have to stay orthogonal so they will move as well by the angle maintaining |v|. That's why I chose the most convenient vectors to work with.

Another reason for doing the problem in general is that your specific vectors might not cover every possibility, which has indeed happened here.

 

[quietrain]

You do not have 3 independent, mutually orthogonal vectors in 2D. One of the vectors a, b, c is linear combination of the other two.

ehild

its something like this right?

*****B
**** ^
**** |
A<---|--->C

since they are 3 independ orthogonal vectors, how can one of them be a linear combination of the other two?

how does he get sqrt 2 or sqrt 5?

 

[ehild]

They are not three independent vectors.

a=c or a=-c, so either v=2a+b or v=b

The magnitude of v can not be sqrt(2).

ehild

 

[HallsofIvy]

You have already been told that there aren't three independent vectors in two dimensional space! The problem does NOT say that they are independent. In particular the problem, does NOT say that c is orthogonal to a. You are given that b is orthogonal to a and c and they are all of unit length which means that either a= c or a= -c.

If a= c, then v= 2a+ b and, since a and b are unit length and orthogonal, the length of v is $\sqrt{2^2+ 1}= \sqrt{5}$. If a= -c, then v= b which has length 1.

 

[quietrain]

ohh.h... isee... thanks!