Question

I need to find all vectors v that are orthogonal to u. Where u = (2,-1,1)

Answer 1

Let v = (x, y , z)

where v is a vector and x,y,z are scalars

Answer 2

if vector v is orthogonal to vector u, then the dot product of the two vectors should be 0


v dot u = 0

(x,y,z) dot (2,-1,1) = 0

2*x + (-1)*y + 1*z = 0

2x - y + z = 0

Answer 3

you can then solve that for z to get

2x - y + z = 0

-y + z = -2x

z = -2x + y

Answer 4

okay, thank you that makes sense, but what do i do from here?
do I also need to solve for x and y?

Answer 5

no, you now let

x = s
y = t

where x,y,s,t are all scalars. This allows you to parameterize things

so

v = (x,y,z)

becomes

v = (s,t,-2s+t)

that's the general vector v that is orthogonal to vector u. To generate a specific vector v (that has numbers instead of variables, you can pick any ordered pair (s,t) and plug them in)

Answer 6

Oh! thank you! that makes sense. That helped a lot thank you for your help!

Answer 7

you're welcome

Answer 8

I'm glad it's making sense now