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If $f;and;vec G$ are point functions prove that the components of
$vec G$ normal and tangential to the surface $f=0$ are $frac(vec
G.nabla f)nabla f(nabla f)^2$ and $fracnabla f times(vec G timesnabla f)(nabla f)^2$

Now $nabla f$ is normal to $f$ and $(vec G .nabla f)nabla f$ projection $G$ on $nabla f$ therefore unit normal would be given by $frac(vec
G.nabla f)nabla fnabla f$ but the question is saying that the denominator would have a square term , I got a similar result for the tangential component also

Please help : And also can someone explian the geometric significance of this problem , as to what is happening between the surfaces.

The vector $nabla f$ is normal to the surface

$f = 0, tag 1$

but is not in general orthonormal to it, being as we typically will not have

$vert nabla f vert = 1 ; texton ; f = 0; tag 2$

nevertheless, wherever

$nabla f ne 0, tag 3$

we may define the unit vector field

$dfracnabla fvert nabla f vert tag 4$

which is orthonormal to $f = 0$, since

$left vert dfracnabla fvert nabla f vert right vert = dfracvert nabla f vertvert nabla f vert = 1; tag 5$

it follows that $vec G^bot$, the component of $vec G$ normal to $f = 0$, is given by

$vec G^bot = left ( vec G cdot dfracnabla fvert nabla f vert right ) dfracnabla fvert nabla f vert = dfrac left ( vec G cdot nabla f right ) nabla fvert nabla f vert^2 . tag 6$

From this equation we see that the component of $vec G$ parallel to $f = 0$ is

$vec G^parallel = vec G - vec G^bot = vec G -dfrac left ( vec G cdot nabla f right ) nabla fvert nabla f vert^2 =
left ( dfracnabla fvert nabla f vert cdot dfracnabla fvert nabla f vert right ) vec G - left ( vec G cdot dfracnabla fvert nabla f vert right ) dfracnabla fvert nabla f vert . tag 7$

We now apply the well-known vector identity

$vec A times (vec B times vec C) = (vec A cdot vec C) vec B - (vec A cdot vec B)vec C tag 8$

to the right-hand side of (7), yielding

$vec G^parallel = left ( dfracnabla fvert nabla f vert right ) times left (vec G times dfracnabla fvert nabla f vert right ) = dfracnabla f times (vec G times nabla f)vert nabla f vert ^2, tag 9$

as desired.