Question

The foot of the perpendicular from the origin to the join of A(-9,4,5) and (11, 0, -1) is?

a) (0, 0, 0)
b) (-2, 2, 3)
c) (2, 0, -1)
d) (1, 2, -2)

Answer 1

The foot of the perpendicular from the origin to the line joining points A(-9,4,5) and B(11,0,-1) is approximately (-9.23, 3.85, 4.86).

Step-by-step explanation:

To find the foot of the perpendicular from the origin to the line joining points A(-9,4,5) and B(11,0,-1), we can use the formula for finding the foot of the perpendicular from a point to a line in three-dimensional space.

First, we need to find the direction vector of the line AB by subtracting the coordinates of point A from the coordinates of point B: AB = B - A = (11,0,-1) - (-9,4,5) = (20,-4,-6).

Then, we can find the position vector of the foot of the perpendicular by projecting the vector AO (where O is the origin) onto the vector AB and adding the result to the position vector of point A:

Foot of perpendicular = A + (AO · AB / |AB|^2) * AB.

Calculating the dot product AO · AB = (-9,-4,-5) · (20,-4,-6) = 242,

Calculating the magnitude of AB: |AB| = sqrt(20^2 + (-4)^2 + (-6)^2) = sqrt(416).

Finally, plugging in the values, we get:

Foot of perpendicular = (-9,4,5) + (242 / 416) * (20,-4,-6) = (-9,4,5) + (121/208) * (20,-4,-6) = (-9,4,5) + (121/208) * (40/26,-8/26,-12/26) = (-9,4,5) + (55/104,-11/26,-33/104) ≈ (-9.23, 3.85, 4.86).