# How to scale an unit vector $u$ in such way that $a u\cdot u=1$ where $a$ is a scalar

I have a problem of scaling a unit vector $u$ in such way that its scalar product (by itself) multiplied by a costant has to be equal to one. $$a u\cdot u=1$$ How do I do it?

Edit: the unit vector - it is a normalized vector which has the length equal to $1$: $||u||=1$

Edit2: I need to rescale this vector $u$ in such way that $a u\cdot u=1$

• What's the definition of a unit vector? – lulu Jul 23 '15 at 13:57
• $$|u|=1\\au.u=1\\a |u| |u|cos (0^0) =1\\a*1*1*1=1$$ – Khosrotash Jul 23 '15 at 14:04
• But if we already have $u \cdot u = 1$ then we can just take a = 1, no? I suspect you meant to start with a vector that was not a unit vector, though I am not certain. – lulu Jul 23 '15 at 14:08
• I see the edits but what you are writing is not at all clear. The equation you wrote, $a\,u \cdot u = 1$ has the unique solution $a = 1$. I now believe you meant to write: "Given a unit vector $u$ and an arbitrary non-zero scalar $a$ find a scalar $\lambda$ such that $a\, (\lambda u) \cdot (\lambda u) = 1$.". Is this correct? But then we see that this has no solution if $a < 0$, trusting everything in sight is real, and if $a > 0$ then we just take $\lambda = \frac {1}{\sqrt {a}}$. Or did you mean something else? – lulu Jul 23 '15 at 14:14
• $a u\cdot u=a(u\cdot u)=a||u||^2=1$ Since $||u||=1$, $a=1$. – user137035 Jul 23 '15 at 14:18

I think you mean that $u$ is a unit vector. If so, $u\cdot u = 1$, and $au\cdot u = 1$ is not possible in general.
But I think what what the question is asking is that you are given $u$ and you are supposed to find $v$ in the same direction as $u$ so that $$av\cdot v = 1.$$ Is that right?
If so, write $v=bu$ and then solve $$a(bu)\cdot (bu)=1$$ to find $b$, using the fact that $u$ is a unit vector so $u\cdot u = 1$.
You can divide vector $v$ by its own modulus and by $\sqrt a$ so that the dot product is 1