Find endpoint of two vectors if they are perpendicular.

Question

The question states: If A = (10,7) and B = (0, -8), find point D, which is on the x-axis, so that the vector AD is perpendicular to the vector BD.

The only information I have is that if two vectors are perpendicular, their dot product is 0. How can I find point D?

Answer 1

Let the point D = (x, 0).

Then vector DA = < 10-x, 7-0 > and vector DB = <0-x, -8-0>

The dot product of those two vectors is: -x(10-x) + 7(-8).

Set that expression equal to zero and solve for x (two solutions).

Note you could also have found expressions for the slope of line AD and line BD. Then by setting up the equation that the product of those slopes is -1 you achieve the same equation and results.