$\def\vec{\overrightarrow}$The full question reads as following:
Prove that if two vectors $\vec{u}$ and $\vec{v}$ in $\mathbb{R}^2$ are orthogonal to a nonzero vector $\vec{w}$ in $\mathbb{R}^2$, then $\vec{u}$ and $\vec{v}$ are scalar multiples of each other.
The entirety of this chapter has been finding an arbitrary point provided we know a point. Naturally I would think that they imbed this theory in the exercises.
Initially I wanted to apply the formula for finding a point $x$, which is
$$x= x_{0} + su + tv.$$
However this is does't seem to get me anywhere.
Any hints to how I could prove this?
Best Answer
We first take note that $w \neq 0.$ If either $u$ or $v$ equal $0$, they must be scalar multiples of each other.
We prove for when $u$ and $v$ are not zero vectors, they are scalar multiples of each other.
Let $u = \binom{u_1}{v_1}, v = \binom{v_1}{v_2},$ and $w = \binom{w_1}{w_2}.$ Since $u$ and $v$ are both orthogonal to $w,$ we have $u \cdot w = 0$ and $v \cdot w = 0.$
In other words,
$$u_1w_1 + u_2w_2 = 0$$ $$\text{and}$$ $$v_1w_1 + v_2w_2 = 0.$$
Rearranging both equations gives us $\frac{u_1}{u_2} = -\frac{w_2}{w_1} = \frac{v_1}{v_2}.$ Therefore, for a value $k \in \mathbb{R}$ we have $u_1 = kv_1$ and $u_2 = kv_2$.
$u$ and $v$ are scalar multiples of each other.