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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.

by Platinum
(106,910 points)
in Mathematics
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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.
by Platinum
(106,910 points)

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