# Orthogonal vectors

• Apr 24th 2011, 12:01 AM
Ife
Orthogonal vectors
I am sure there is an easy way to reply to this. however I am getting a bit confused by this:

Let v= the vector (a, b) (I can't input the matrix array tonight for some reason). Describe the set H of vectors (x,y) that are orthogonal to v. [Hint: Consider v=0, and v not = 0]
• Apr 24th 2011, 12:16 AM
abhishekkgp
Quote:

Originally Posted by Ife
I am sure there is an easy way to reply to this. however I am getting a bit confused by this:

Let v= the vector (a, b) (I can't input the matrix array tonight for some reason). Describe the set H of vectors (x,y) that are orthogonal to v. [Hint: Consider v=0, and v not = 0]

let v not= 0.
then (x,y) is orthogonal to (a,b) if ax+by=0 (dot product is zero). thats an equation of a straight line.
• Apr 24th 2011, 12:21 AM
Ife
Quote:

Originally Posted by abhishekkgp
let v not= 0.
then (x,y) is orthogonal to (a,b) if ax+by=0 (dot product is zero). thats an equation of a straight line.

Thanks, now I was thinking its a straight line, but I wasn't sure exactly how I should lay it out. But the problem is also that this question i see was worth 6 marks.. is that sufficient explanation for a 6 marker? I was thinking there should be more?
• Apr 24th 2011, 12:26 AM
Deveno
describing the set when v = 0 is also important.....
• Apr 24th 2011, 12:29 AM
abhishekkgp
Quote:

Originally Posted by Ife
Thanks, now I was thinking its a straight line, but I wasn't sure exactly how I should lay it out. But the problem is also that this question i see was worth 6 marks.. is that sufficient explanation for a 6 marker? I was thinking there should be more?

if it is a six marker you may be expected to be very fastidious. Now i am not sure if (0,0) is considered to be orthogonal to a non-zero vector although its dot prod with any other vector will be zero. If not, then from the points of the straight line you have to exclude out the origin. If yes then party anyway.
also see denevo's post.
• Apr 24th 2011, 12:32 AM
Ife
hmm, a bit confused once again..
• Apr 24th 2011, 01:16 AM
Deveno
the cases v = 0, and v ≠ 0 produce two very different kinds of sets for H. this is important.

if v = (a,b) ≠ (0,0), then the space span{v} is a line. think about what it means for a vector to be perpendicular (orthogonal) to a line.

however, if v = (0,0), then span{v} is just the origin. every vector is by definition perpendicular (orthogonal) to the origin. look:

(x,y).(0,0) = x*0 + y *0 = 0 + 0 = 0.