Linear Algebra - Find unit vector orthogonal to 2, 4-space vectors?

[twotaileddemon]

Homework Statement

Given the vectors
u = (2, 0, 1, -4)
v = (2, 3, 0, 1)
Find any unit vector orthogonal to both of them

Homework Equations

I know that two vectors are orthogonal if their dot product is zero...

The Attempt at a Solution

I don't even know how to begin! I know the unit vector for one vector is the vector over its magnitude, but what about two of them? How do I find something that's orthogonal to both at the same time?

Please help ASAP! :) Thanks.

 

[Mark44]

Hint: Gram-Schmidt

 

[twotaileddemon]

I haven't learned that in class, but I did wikipedia it..
according to that, the answer wouldn't change since <u,v> = 0 and the GS method just uses u - proj_vu which would be <u,v>v/<v,v> with a 0 on top... giving only (2, 0, 1, -4) which would not work.

any other suggestions please? or advice? Thank you :)

edit: please consider helping me with this, even if just a tiny tiny bit of guidance. I have been trying to figure this out for hours.. it's over 2 am my time.. and I'm really tired :/

I will work hard and along with you! I need to understand this problem to do other problems in the HW too...

we haven't learned the method you mentioned in class... we have been talking about euclidean inner product and vectors in R^n space

 

[angryfaceofdr]

So you know you are looking for a unit vector, call it $\vec{a}=<a_1,a_2,a_3,a_4>$. Since $\vec{a}$ is a unit vector we must have $|\vec{a}|=a_1^2+a_2^2+a_3^2+a_4^2=1$.

What else do you know?

 

[twotaileddemon]

So you know you are looking for a unit vector, call it $\vec{a}=<a_1,a_2,a_3,a_4>$. Since $\vec{a}$ is a unit vector we must have $|\vec{a}|=a_1^2+a_2^2+a_3^2+a_4^2=1$.

What else do you know?

that 2a_1 + a_3 - 4a_4 = 0
and 2a_1 + 3a_2 + a_4 = 0

 

[angryfaceofdr]

True, what can you do with these three equations?

 

[twotaileddemon]

put them in matrix form and try to eliminate one of the variables?

 

[angryfaceofdr]

See what happens

 

[twotaileddemon]

See what happens

I did that before.. but whenever I would eliminate one of the variables, another would "return"

like if I put it in the form

[2 0 1 -4 | 0]
[2 3 0 1 | 0]

lets say I subtracted row one (R1) from row two (R2) ... R2 - R1
then I get

[2 0 1 -4 | 0]
[0 3 -1 5 | 0]
.. which makes me lose a variable but gain another in the process.

this is where I got stuck before

 

[angryfaceofdr]

So we either need one more equation or to get 'rid' of one variable

 

[angryfaceofdr]

Suppose, $\vec{a}=<b,c,d,0>$. How many equations and unkowns do we have now?

 

[twotaileddemon]

I don't understand what I could eliminate or add though...

unless you mean solving for something like a_1 to get it in terms of the other three? but that seems kinda like we're I'm stuck at now...

 

[angryfaceofdr]

i mean, set a_4=0

 

[twotaileddemon]

Suppose, $\vec{a}=<b,c,d,0>$. How many equations and unkowns do we have now?

If that is the case, then we have:

2b + d = 0
2b + 3c = 0
b + c + d = 1

...correct?

so three equations and 3 unknowns?
which I think I can actually solve...

is this allowable? and why so?

 

[angryfaceofdr]

Yes that's right. Well, the problem states to find any unit vector orthogonal to u and v

 

[angryfaceofdr]

And I guessed that we needed 4 equations and 4 unkowns or 3 eqns and 3 unknowns

 

[twotaileddemon]

that's true.. so hypothetically, if I wanted to, I could make another variable 0?

I'm not going to in this case because I can solve the problem fine with the method you gave me.. but in the future I am allowed to put as many 0's as I'd like? Or is there some kind of limit?

thank you so much for your help and patience!

 

[angryfaceofdr]

Try it, and see if you get a suitable vector (unit length orthogonal to both)

 

[angryfaceofdr]

To tell you the truth, I don't know... I think you might lose too much information if you use too many zeros

 

[angryfaceofdr]

I usually just eliminate enough variables so that it fits with the number of equations i have (and pray it works)

 

[twotaileddemon]

ok, I will try to finish the problem and let you know how it goes out!
even if it's not right, you took time to help me and that's really great! you are very nice and I'm sure that I can find a viable solution thanks to you.

 

[angryfaceofdr]

b + c + d = 1

...correct?

whooops should be b^2+c^2+d^2=1

 

[angryfaceofdr]

no problem, good luck!




HINT:

Spoiler

I get two solutions: <3/7,-2/7,-6/7,0>, <-3/7,2/7,6/7,0>

 

[twotaileddemon]

I got

(1/3, -2/3, -2/3, 0) and (-1/3, 2/3, 2/3, 0) as my answers and checked the dot product with but U and V and got 0...

 

[angryfaceofdr]

which is what you want right ?

 

[angryfaceofdr]

means that the two vectors are orthogonal to each other

 

[twotaileddemon]

wait mine only works for u not v.. let me try again.

 

[angryfaceofdr]

hmmm... so
2b+d=0 ----> d=-2b
2b+3c=0----> c=-(2/3)bb^2+c^2+d^2=1=(4/9)b^2+4b^2+b^2

b^2=9/49

 

[twotaileddemon]

so b = +/- 7/3, d = +/- 14/3 and c = +/- 14/9

I think this one works.. just checked all possible 4 scenarios for u and v and got 0

thanks a lot!

 

[angryfaceofdr]

errr... I got b=+/- 3/7 = $$\pm \frac{3}{7}$$

ahh okay cool

 

[twotaileddemon]

yes yes, you are right once again!

I'm sorry, I make lots of mistakes at 4:40 am xD
I wanted to do this homework so badly so I must get it done!

 

[angryfaceofdr]

its fine, happens to everyone

 

[HallsofIvy]

In order to solve for specific values of a, b, c, and d, you would need four equations. But the problem said "Find any unit vector orthogonal to both of them". Just as, in three dimensions, there are an infinite number of unit vectors orthogonal to a single vector, in four dimensions there are an infinite number of unit vectors orthogonal to two vectors.

In order that (a, b, c, d) be orthogonal to u = (2, 0, 1, -4) and v = (2, 3, 0, 1) we must have, as you say, 2a+ c- 4d= 0 and 2a+ 3b+ d= 0. Subtracting the first equation from the second, 3b- c+ 5d= 0 so c= 3b+ 5d. Put that back into either of the first two equations and you can solve for a in terms of b and d. That gives every vector that is orthogonal to u and v in terms of b and d.