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 $\frac{({x}^{2}\hat{i}+{y}^{2}\hat{j}+{z}^{2}\hat{k})}{a}$

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 $\frac{({x}^{2}\hat{i}+{y}^{2}\hat{j}+{z}^{2}\hat{k})}{a}$