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Let $ w= (\sin\theta, \cos\theta)$ and let $w^p$ be a vector perpendicular to $w$ on $\Bbb R^2 $. Where $\Bbb R$ is the real number system. How to show any line in $\Bbb R^2$ is of the form $L={tw+sw^p, s\in \mathbb{R}}$.

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Rotate $w$ and $w^p$ by $-\theta$. Then $w$ and $w^p$ become unit vectors along the $x$- and $y$-axes respectively. So your $L$ has reduced to $t x + s y$, or, using more familiar notation, $a x + b y$.

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