Question

Show that every normal line to th sphere x^2+y^2+z^2 = r^2 where r is the radius passes through the center of the sphere.

Answer 1

take gradient of the surface

Answer 2

ok. (2x,2y,2z) is it?

Answer 3

yeah ... the normal is
x - x1 .....
----- = --------
2x_1 .....

Answer 4

put x_1, y_1, z_1 in place of x, y, z ... assume it to be point on surface of sphere.

Answer 5

i need guidance

Answer 6

or in parametric form
\[ r(t) = (x_1.y_1,z_1)+t(2x_1.2y_1,2z_1) \]
for t=-1/2 ... this always pass through (0,0,0)

Answer 7

or just put the line in this form