I must show that any vectors $\vec{w}$ in $\Bbb R^2$ can be written as a linear combination of two non-zero orthogonal vectors $\vec{u}, \vec{v} \in \Bbb R^2$ using basic geometry concepts such as the dot product.
So I must prove that $\vec{w} = \alpha \vec{u} + \beta \vec{v}$, with $\alpha, \beta \in \Bbb R$.
The thing is that I don't understand why the two vectors must be orthogonal because if I have :
$\vec{u} = \begin{pmatrix} u_1 \\
u_2 \end{pmatrix}$ and
$\vec{v} = \begin{pmatrix} v_1 \\
v_2 \end{pmatrix}$, two non-zero vectors (let's say $u_1$ and $v_2$ can't be equal to zero).
And I say that $\alpha_1 = -\dfrac{v_2}{-u_1v_2 + u_2v_1}$ and $\beta_1 = \dfrac{u_2}{-u_1v_2 + u_2v_1}$.
I find $\vec{w_1} = \alpha_1 \vec{u} + \beta_1 \vec{v} = \begin{pmatrix} 1 \\
0 \end{pmatrix}$.
With $\alpha_2 = -\dfrac{v_1}{-u_2v_1 + u_1v_2}$ and $\beta_2 = \dfrac{u_1}{-u_2v_1 + u_1v_2}$, I find $\vec{w_2} = \alpha_2 \vec{u} + \beta_2 \vec{v} = \begin{pmatrix} 0 \\
1 \end{pmatrix}$.
Therefore with $\vec{u}$ and $\vec{v}$, I can generate $\Bbb R^2$.
Then why would they need to be orthogonal ? Did I miss something or did I do something wrong ? Or maybe I am right ?
Best Answer
You don't need orthogonality per se, only the fact that $u_{1}v_{2} - u_{2} v_{1} \neq 0$, namely that the vectors are non-proportional, or that they form a linearly independent set.
Your author (or instructor) is setting you an easier task: If $u$ and $v$ are orthogonal non-zero vectors in the plane, it's easy to show (explicitly using the dot product, without decomposing vectors into Cartesian components) every vector in the plane is a linear combination of $u$ and $v$. Presumably you'll soon generalize, obtaining the same conclusion under less stringent hypotheses.