# 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.''

I have a question about the solution to this problem ''find the flux of $x\stackrel{^}{i}+y\stackrel{^}{j}+z\stackrel{^}{k}$ through the sphere of radius a and center at the origin. Take n pointing outward.''
The answer in the book was, we have $n=\frac{\left(x\stackrel{^}{i}+y\stackrel{^}{j}+z\stackrel{^}{k}\right)}{a}$; 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 $\frac{\left({x}^{2}\stackrel{^}{i}+{y}^{2}\stackrel{^}{j}+{z}^{2}\stackrel{^}{k}\right)}{a}$
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silenthunter440
A dot product is a scalar but the expression you guessed is a vector.
Ponder
$\left(x\stackrel{^}{i}+y\stackrel{^}{j}+z\stackrel{^}{k}\right)\left(x\stackrel{^}{i}+y\stackrel{^}{j}+z\stackrel{^}{k}\right)={x}^{2}\stackrel{^}{i}\stackrel{^}{i}+xy\stackrel{^}{i}\stackrel{^}{j}+xz\stackrel{^}{i}\stackrel{^}{k}+yx\stackrel{^}{j}\stackrel{^}{i}+\cdots ={x}^{2}+{y}^{2}+{z}^{2}.$
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Nigro6f
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
$F\cdot n={F}^{1}{n}^{1}+{F}^{2}{n}^{2}+{F}^{3}{n}^{3}=\frac{{x}^{2}+{y}^{2}+{z}^{2}}{a},$
and since you are on a sphere ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2}$, obtaining $F\cdot n=a$, as wished.