I don't understand the solution of this problem

Question

I have a question about the solution to this problem ''find the flux of $x\hat i + y \hat j + z \hat k$ through the sphere of radius $a$ and center at the origin. Take $n$ pointing outward.''

The answer in the book was, we have $n =\frac{(x \hat i + y \hat j + z \hat k)}{a} $; therefore $F .n = a$ and then they integrate it, but what I don't get is how $F.n=a$ isn't the vector $n$ the same vector as $F$ but scaled by $1/a$ so the dot product must be $\frac{(x^2 \hat i +y^2 \hat j +z^2 \hat k )}{a} $

Answer 1

Welcome opaque. The dot product $F\cdot n$ gives you a scalar which represent the length of the projection of $F$ on the normal outer vector $n$. This means that by definition $$ F\cdot n=F^1n^1+F^2n^2+F^3n^3=\frac{x^2+y^2+z^2}{a}, $$ and since you are on a sphere $x^2+y^2+z^2=a^2$, obtaining $F\cdot n=a$, as wished.

Answer 2

A dot product is a scalar but the expression you guessed is a vector.

Ponder

$$(x\hat i+y\hat j+z\hat k)(x\hat i+y\hat j+z\hat k)=x^2\hat i\hat i+xy\hat i\hat j+xz\hat i\hat k+yx\hat j\hat i+\cdots=x^2+y^2+z^2.$$