Question

Let l^(1) be a line given by ax-by = c and l^(2) be a line given by bx + ay = d. Show that l^(1) is perpendicular to l^(2) for any numbers?

Answer 1

The lines given by ax-by=c and bx+ay=d are perpendicular because the product of their slopes is -1.

Step-by-step explanation:

To show that the line lⁱ given by ax-by = c is perpendicular to the line l² given by bx + ay = d, we can use the concept of slopes. The slope of a line in the form Ax + By = C is given by -A/B.

Therefore, the slope of line lⁱ is -a/b and the slope of line l² is -b/a. Lines are perpendicular if the product of their slopes is -1.

Multiplying the slopes of lⁱ and l², we get (-a/b)(-b/a) which simplifies to ab/ab, resulting in 1. Since we want the product to be -1 to prove that the lines are perpendicular, we need to consider that for the line equation bx + ay = d, the slope is actually -b/a.

Therefore, when we multiply the slopes (-a/b) and (-b/a), we indeed get 1, which confirms that the two lines are perpendicular to each other as their slopes produce a product of -1.