Suppose $\bf p, q, r$ are three non-coplanar vectors in ${\mathbb{R^3}}$. There is a vector $\bf x$ having projections along them are $a, b$ and $c$ respectively. Then can we write $$ \bf x = a\bf p + b\bf q + c\bf r $$? If yes, why?
I think this cannot be done unless all three vectors are mutually perpendicular unit vectors.
No, it is not true. Take$$\mathbf p=\left(1,1,0\right),\ \mathbf q=\left(1,0,1\right)\text{, and }\mathbf r=\left(0,1,1\right)$$and $\mathbf x=(1,1,1)$. Then $a=b=c=1$, but $a\mathbf p+b\mathbf q+c\mathbf q=(2,2,2)\neq\mathbf x$.
Yes, if you want that the equality holds, then the vectors $\mathbf p$, $\mathbf q$, and $\mathbf r$ should be orthogonal.