Calculus 3 slopes

Question

Show that v = ai +bj is perpendicular to the line ax + by = c by establishing that the slope of the vector v is the negative reciprocal of the slope of the given line.

Answer 1

Part A) First, we determine the slope of the line ax+by=c by expressing the line in slope-intercept form.

1. by=-ax+c [subtract ax from both sides]
2. y=-(a/b)x+(c/b) [divide both sides by b to get line in slope-intercept form]
3. -a/b is the slope of the line!

Part B) Next, we determine the slope of the vector v=ai+bj

1. The slope of a vector in 2d cartesian space is given by the vector's vertical component (b) divided by the horizontal component (a), much like how the slope of a line is Δy/Δx.
2. We see that the slope of the vector is b/a

Part C) Verify that the vector is perpendicular to the line by establishing that its slope is equal to the negative reciprocal of the line's slope.

1. Let m be the slope of the line found in Part A. Then, m=-a/b
2. Taking the reciprocal, 1/m=-b/a
3. Multiply the reciprocal by -1, -1/m=b/a
4. b/a is precisely the slope of the vector found in Part B