Use the dot product to show that the line c = by + ax is perpendicular to vector v = <a,b>

Question

They seem like they would be parallel. doesn't c also = <a,b>?

Answer 1

changing c = by + ax to the form y = mx + b gives us

ax + by = c
by = -ax + c
y = (-a/b)x + c/b

so the slope of the line is -a/b
or a vector in the direction of the line is (b,-a)
(remember, slope is rise/run, so -a/b has a run of b and a rise of -a, or direction vector (b,-a))

then (b,-a).(a,b) = ab - ab = 0
so the line is perpendicular to the given vector.