Thread: Linear Algebra "show that" problem

Show that the lines given by the equation ax + by + c = 0 and bx - ay + d = 0 (where a, b, c, d are in R) are perpendicular by finding a vector in the direction of each line and showing that these vectors are orthogonal. (Hint: Watch out for the cases in which a or b equals zero.)

I am not sure how to begin this problem...if someone could help, I would greatly appreciate it. Thanks!

Do you know how to perform the cross product?

Yes, but I don't think I should do that for this proof because I learned it in another class. This class is quite rigorous, proving everything step-by-step.

What I don't understand is how to obtain vectors from the equations. I know how to create a specific vector by plugging in numbers for a,b, and c; ex:
a=1 b=1 c=-2

However, based on the hint I think it needs to be more general, but I'm not sure. So really I don't know what to do.

Originally Posted by paulrb

Show that the lines given by the equation ax + by + c = 0 and bx - ay + d = 0 (where a, b, c, d are in R) are perpendicular by finding a vector in the direction of each line and showing that these vectors are orthogonal. (Hint: Watch out for the cases in which a or b equals zero.)

I am not sure how to begin this problem...if someone could help, I would greatly appreciate it. Thanks!

for perpendicular: you want the dot product of the vectors to be zero

for parallel: you want the cross product of the vectors to be zero.

the slope can help you find the vector in the direction of the line. slope = rise/run = (y-component of the vector)/(x-component of the vector)


as far as not being sure whether you can use dot and/or cross products or not, we can't help you there. if your professor asked you this question he would reasonably give you methods of what he would accept as proofs of these facts, or assigned readings that would do that