Among all unit vectors $\vec{u}=\scriptsize\begin{bmatrix}x\\y\\z\end{bmatrix} \in \mathbb{R}^3$ which one minimizes the sum $x+2y+3z$?

Question

My first instinct, of course, was to use Lagrange multipliers, but I have to use linear algebra to solve this.

How would I construct an orthonormal basis in this case? I'm not sure how to parametrize the span of vectors. Should I use Cauchy-Schwarz, or the fact that $\theta=\text{arccos}\frac{\vec{x}\cdot\vec{y}}{||\vec{x}||\,||\vec{y}||}$?

Answer 1

hint: $$(x+2y+3z)^2 \leq (1^2+2^2+3^2)(x^2+y^2+z^2)$$ by CS inequality.