# Find a normal vector to a unit sphere using cartesian coordinates

1. Oct 8, 2015

### Frozen Light

1. The problem statement, all variables and given/known data
Consider a unit sphere centered at the origin. In terms of the Cartesian unit vectors i, j and k, find the unit normal vector on the surface

2. Relevant equations
A dot B = AB cos(theta)
A cross B = AB (normal vector) sin(theta)

3. The attempt at a solution
Isn't any direction a normal vector?
i x j = + k
j x i = - k
etc.

2. Oct 8, 2015

### Staff: Mentor

Any nonzero vector would be a normal at some point on the surface of the sphere. My guess is that you should take an arbitrary point P(x, y, z) on the surface, and find the normal to it. If that's what is wanted in the problem, it could have been written more clearly.

3. Oct 8, 2015

### Frozen Light

Thank you, that would make a bit more sense.

4. Oct 9, 2015

### HallsofIvy

Staff Emeritus
The unit sphere is of the form $x^2+ y^2+ z^2= 1$. You can think of that as a 'Level Surface" of the function $F(x, y, z)= x^2+ y^2+ z^2$ and use the fact that the gradient of such a function, $\nabla F$, is always normal to level surfaces.

5. Oct 9, 2015

### LCKurtz

Or think about what direction a position vector to a point on the sphere has.