Question:
Let $\vec{u}_1, \dots, \vec{u}_k$ be pairwise orthogonal unit vectors in $\mathbb{R}^n$. Prove that $$ \vec{c}^\mathsf{T} \vec{c} \geq \sum_{i=1}^k(\vec{c}^\mathsf{T} \vec{u}_i)^2 $$ for all $\vec{c} \in \mathbb{R}^n$.
My attempt is the following:
Define $\vec{c} = [c_1 \dots c_n]^\mathsf{T}$and rewrite $$ \vec{c}^\mathsf{T} \vec{c} = \sum_{i=1}^n c_i^2 $$ Also, if $\vec{u}_1, \dots, \vec{u}_k$ are pairwise orthogonal unit vectors in $\mathbb{R}^n$, we have $k \leq n$ (this seems to be true, but I don't know the reference). Therefore, to show the above inequality, it suffices to show that $$ c_i \geq \vec{c}^\mathsf{T} \vec{u}_i $$ for all $i = 1, \dots, k$. I don't what should do next and how to apply the property of pairwise orthogonal unit vectors. Any help is appreciated.