Question

​​​​​​​Let x be a point and let L be a line. Prove that there is a unique line m through X perpendicular to L. Fur thermore, Prove that m=X+[N] where N is a unit normal vector L.

Answer 1

To prove that there is a unique line m through point X perpendicular to line L, we need to show that such a line exists and that it is the only one.

Step-by-step explanation:

To prove that there is a unique line m through point X perpendicular to line L, we need to show that such a line exists and that it is the only one.

First, we know that if a line is perpendicular to another line, their slopes are negative reciprocals of each other. So, let L have slope 'm1', then the slope of the line m perpendicular to L will be -1/m1.

Next, to find the equation of line m, we can use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the coordinates of point X. Substitute the slope -1/m1 and the coordinates of X into the equation to get the equation of line m. This equation will be unique for a given point X and line L.