Calculating dot product between a vector and its component

Question

Let $\hat u, \hat v$ and $\hat w$ be unit orthogonal vectors, and given $\vec a = \alpha \hat u +\beta \hat v + \gamma \hat w$ prove that $\alpha = \vec a \cdot \hat u$

I tried using $ \vec a \cdot \hat u = \|\vec a \| \|\hat u \| \cos \theta$, but I don't know $\|\vec a\|$ and $\cos \theta $.

Then, I tried using $ \vec a \cdot \hat u = (\alpha, \beta, \gamma)\cdot(u_1,u_2,u_3)$ but I don't know $\vec u$ components, just its module.

If I use $\hat u= \hat i =(1,0,0)$ I get the answer easily, but this is not the case...

Answer 1

Try $$\vec{a}\cdot\hat{u}=(\alpha\hat{u}+\beta\hat{v}+\gamma\hat{w})\cdot\hat{u}= \alpha\hat{u}\cdot\hat{u}+\beta\hat{v}\cdot\hat{u}+\gamma\hat{w}\cdot\hat{u} =\alpha\cdot 1+0+0=\alpha. $$ This works, because $\hat{u}, \hat{v},$ and $\hat{w}$ are unit orthogonal vectors.