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(a) The equation $a x+b y=0$ represents a line through the origin in $R^{2}$. Show that the vector $\mathbf{n}_{1}=(a, b)$ formed from the coefficients of the equation is orthogonal to the line, that is, orthogonal to every vector along the line.

(b) The equation $a x+b y+c z=0$ represents a plane through the origin in $R^{3}$. Show that the vector $\mathbf{n}_{2}=(a, b, c)$ formed from the coefficients of the equation is orthogonal to the plane, that is, orthogonal to every vector that lies in the plane.
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We will solve both problems together. The two equations can be written as $$(a, b) \cdot(x, y)=0 \text { and }(a, b, c) \cdot(x, y, z)=0$$ or, alternatively, as $$\mathbf{n}_{1} \cdot(x, y)=0 \quad \text { and } \quad \mathbf{n}_{2} \cdot(x, y, z)=0$$ These equations show that $\mathbf{n}_{1}$ is orthogonal to every vector $(x, y)$ on the line and that $\mathbf{n}_{2}$ is orthogonal to every vector $(x, y, z)$ in the plane.
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