Determining the general form of an orthogonal vector

[MoreDrinks]

Homework Statement

Determine a vector that is orthogonal to both (1,2,-1) and (3,1,0)

Homework Equations

As above.

The Attempt at a Solution

The solution, from the back of the book, is "any vector of the form (a, -3a, -5a), but I'm not sure how they got there. I get the methodology for a matrix in two dimensions, like so:

Find a matrix orthogonal to (5,1)

(5a) + (-b) = 0
5a = b
The answer is (a, 5a) or any vector of the form a(1,3)

...but I don't understand how to go about this for the question originally stated.

 

[STEMucator]

Homework Statement

Determine a vector that is orthogonal to both (1,2,-1) and (3,1,0)

Homework Equations

As above.

The Attempt at a Solution

The solution, from the back of the book, is "any vector of the form (a, -3a, -5a), but I'm not sure how they got there. I get the methodology for a matrix in two dimensions, like so:

Find a matrix orthogonal to (5,1)

(5a) + (-b) = 0
5a = b
The answer is (a, 5a) or any vector of the form a(1,3)

...but I don't understand how to go about this for the question originally stated.

Are you familiar with inner products at all? They will be very helpful to you for this problem.

Two vectors, say u and v, are orthogonal when their inner product is zero. That is when <u,v> = 0.

 

[MoreDrinks]

Are you familiar with inner products at all? They will be very helpful to you for this problem.

Two vectors, say u and v, are orthogonal when their inner product is zero. That is when <u,v> = 0.

If that means the same thing as dot product, yes. And I know that two vectors are orthogonal only if that is zero. I also know that the cross product - not taught, just looked it up - will give a vector that is perpendicular/orthogonal to two vectors, but it's not giving me the general answer I'm looking for here...I don't think. If it does, I don't see it.

 

[STEMucator]

If that means the same thing as dot product, yes. And I know that two vectors are orthogonal only if that is zero. I also know that the cross product - not taught, just looked it up - will give a vector that is perpendicular/orthogonal to two vectors, but it's not giving me the general answer I'm looking for here...I don't think. If it does, I don't see it.

Yes the dot product is the same thing. So for the vector you're trying to find to be orthogonal to both of the vectors you've been given, its dot product must be zero with each of them.

Also, the cross product is useful here as well since you're working in $ℝ^3$. If you take the cross product of the two vectors you've been given, you will retrieve a vector which is orthogonal to both of the vectors at the same time.

 

[LCKurtz]

And remember, if you find any vector which works, a constant times it will work too.

 

[MoreDrinks]

Yes the dot product is the same thing. So for the vector you're trying to find to be orthogonal to both of the vectors you've been given, its dot product must be zero with each of them.

Also, the cross product is useful here as well since you're working in $ℝ^3$. If you take the cross product of the two vectors you've been given, you will retrieve a vector which is orthogonal to both of the vectors at the same time.

Thanks for your help. Is there any standard way to find a vector with a dot product that equals zero with both, and/or gives a general result of the form I showed in the OP?

 

[LCKurtz]

Thanks for your help. Is there any standard way to find a vector with a dot product that equals zero with both, and/or gives a general result of the form I showed in the OP?

Yes. If you dot (a,b,c) into your two vectors and set those equal to zero, you will get two equations in three unknowns. That will leave a free variable in your answer.

 

[MoreDrinks]

Yes. If you dot (a,b,c) into your two vectors and set those equal to zero, you will get two equations in three unknowns. That will leave a free variable in your answer.

Well I'll be a monkey's uncle, it worked. Now I feel silly for even asking, but thanks! There will probably be more threads and questions as I go through some vector chapters here.