Calc Problem: Line perpendicular to a vector, proving.?

Question

Show that vector v=ai+bj is perpendicular to line ax+by=c by establishing that slope of v is negative reciprocal of slope of given line. Any help would be much appreciated. Thanks!

Answer 1

two lines are perp if the product of their slopes is -1

if you write the equation of a line in the form:

y=mx+b, then m, the coefficient of x, is the slope of the line

so, the vector v=ai +bj where i and j are unit vectors in the x and y directions...if you draw v, you will see that the slope of this line is b/a (since the slope of a line is equal to the tan of the angle it makes with the x axis)

now, take the second line:

ax+by=c

y=-a/b x +c/b

the slope of this is -a/b

multiply the two slopes:

b/a x -a/b = -1

and the two lines are perp

Answer 2

ax + by = c

by = - ax + c

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

m1 = - a / b

Vector v has slope m2 = b / a

m1 x m2 = (- a / b ) ( b / a ) = - 1

Vector is perpendicular to line.