dot product of a vector in a given direction

Question

I have a vector field $\vec A = (y^3+z^3) \hat i + (x^3+z^3) \hat j + (y^3+x^3) \hat k$ defined over a unit sphere $x^2+y^2+z^2=1$

what is the magnitude of the component $\vec A$ normal to the spherical surface at the point $(1/\sqrt 3,1/\sqrt 3,1/\sqrt 3)$ ?

What i have done is, I have found the unit normal vector at the given point by parameterizing the sphere. then I found the dot product given vector field at the given point with the unit normal vector, which gave me a magnitude 0.66666

The actual answer is 1.333333

where did i go wrong?

Answer 1

The unit normal to the units sphere is just the coordinates to the given point on the sphere. So your dot product becomes:

\begin{align*} \vec A \cdot \vec n = 3((\frac{1}{\sqrt{3}})^3+(\frac{1}{\sqrt{3}})^3)\frac{1}{\sqrt{3}} = 6 (\frac{1}{\sqrt{3}})^4 = \frac{6}{9} = \frac{2}{3}. \end{align*}

So in short - you are right.