# arrow_back 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.

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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.

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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