A question from my physics test (but actually it a question about vectors):
let $t\in [0, T]$ and let $\overrightarrow{v}(t)$ denote the velocity vector in $\mathbb{R}^3$. Assume that $\overrightarrow{v}(t)=(1, 0, 0)$ for any $t\in [0, T]$. What one can say about $\overrightarrow{v}$?
In my opinion the question is pretty vague. Anyway, I answered it means that the velocity is constant in any direction and has null components along $\hat{y}$ and $\hat{z}$ direction. The lecturer said that my answer is not correct since I should have indicate in which halplane/halfspace the velocity vector is placed, give information about magnitude (easy, its magnitude is $1$) and if there is a physical situation which guarantees that $\overrightarrow{v}=(1, 0, 0)$ is in that form.
I don't know how to answer these questions.
Could someone please help? Thank you.
As you say the $\hat{y}$ and $\hat{z}$ componentes are zero, so the velocity "live" along the $x$-line, this is the subspace. If you traslated the point (or body) such that a time $t=0$ their position is the origin $(0,0,0)$ then the point (or body) will be always in the subspace $\{(x,0,0):x > 0$ }$.
For the physical situation, can think in a particle with position $\vec{x}(t)=(t+a,b,c)$ with $a,b,c \in \mathbb{R}$ constant. Then it's velocity is $\vec{v}(t)=\vec{x}'(t)=(1,0,0)$.