How to tell if these lines are perpendicular to each other?

Question

l₁: 7x + 2y - 3 = 0

l₂: l(t) = (1 - t) * Matrix [1 0] + t * Matrix [-6 -2] where both matrices are 2x1.

Thanks!

Answer 1

If t is a scalar quantity then you get a vector (1-7t, -2t) as I_2. You haven't specified what t is.

M1, the slope of I₁ is -7/2 by some algebra.

If we pick t = 0 and t=1, I₂ will have a slope, M2 of 2/7. We generate two points from the line's parametric form and calculate the slope using definition of slope of a line.

M1*M2 = -1

Therefore the lines are perpendicular to each other.

We should be able to write I₁ in vector form also. Then if the dot product of the vector representation of I₁ and I₂ as written is zero, they are orthogonal.

Feel free to ask if there is something that you don't understand.

Answer 2

For I₂, x = 1(1-t) - 6t = 1 - 7t and y = 0(1-t) - 2t = -2t

Since y = -2t, t = (-1/2)y. So, x = 1 - 7(-1/2)y.

(7/2)y = x - 1

y = (2/7)x - 2/7. So, I₂ has slope 2/7.

For I₁, 2y = -7x - 3. So, y = (-7/2)x - 3/2, which has slope -7/2.

Since the slopes of the lines are negative reciprocals of each other, the lines are perpendicular.