I've seen that to find a normal vector to a line such as $3x+4y-1=0$ people take the coefficients and say $(3,4)$ is a normal vector?
Why does this work? How are the coefficients related to the normal?
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$\begingroup$ @ZacharySelk I think its a plane in 3 dimensions. Although normal vector will be <3,4,0> right? $\endgroup$– prog_SAHILJan 8, 2018 at 4:18
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$\begingroup$ @prog_SAHIL They said it's a line but if it was a plane you would be correct. $\endgroup$– user223391Jan 8, 2018 at 4:21
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$\begingroup$ @ZacharySelk I've corrected it $\endgroup$– K.M.Jan 8, 2018 at 4:22
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$\begingroup$ The coefficients, not the constant term though, form a normal vector. $\endgroup$– max_zornJan 8, 2018 at 4:28
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$\begingroup$ Trivially. No pun intended. $\endgroup$– EvpokJan 8, 2018 at 10:45
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6 Answers
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Two vectors $a,b$ are normal iff $a\cdot b=a_1b_1+...+a_2b_2=0$.
If you have a vector, $(x,y)$ and you want to find vectors that are normal to it you want to find a vector $(a,b)$ such that $ax+by=0$. So a normal vector to the line $3x+4y=0$ is simply $(3,4)$.
If you have a constant on the right side, it just moves the line up or down. It doesn't change anything else. So a normal vector will still be $(3,4)$.
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2$\begingroup$ The normal vector is not unique, so you might want to call this "a normal vector"? $\endgroup$– max_zornJan 8, 2018 at 4:29
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$\begingroup$ @max_zorn sure. $\endgroup$– user223391Jan 8, 2018 at 4:30
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$\begingroup$ sorry I don't know If I'm being retarded but I don't get how (3,4) is a normal vector to the line 3x+4y=0 $\endgroup$– K.M.Jan 8, 2018 at 4:35
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3$\begingroup$ @K.M. the vector $(3,4)$ is orthogonal to any vector $(x,y)$ which is on that line because $$(3,4)\cdot (x,y)=3x+4y=0$$ $\endgroup$– DaveJan 8, 2018 at 5:20
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1$\begingroup$ @K.M. To be more specific, $(3,4)$ is normal to any vector pointing along the line $3x + 4y + c = 0$ for any constant $c$. If $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line, then $(x_2 - x_1, y_2 - y_1)$ is a vector pointing along the line. Subtracting $3x_1 + 4y_1 + c = 0$ from $3x_2 + 4y_1 + c = 0$, you get $$3(x_2 - x_1) + 4(y_2 - y_1) = 0$$which tells you that $(3,4)$ is normal to the vector $(x_2 - x_1, y_2 - y_1)$. $\endgroup$ Jan 8, 2018 at 14:39
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If the line equation is $$ ax+by+c=0$$
then, the normal vector is $\vec{n}=\left(\begin{array}{c}a \\ b\end{array}\right)$, and the direction vector is $\vec{v}=\left(\begin{array}{c}-b \\ a\end{array}\right)$
Demonstration:
First, we begin by showing that $\vec{n}=\left(\begin{array}{c}a \\ b\end{array}\right)$
It is easy to see that if $a=(x_a,y_a)$, $b=(x_b,y_b)$ are two points from the given line then, $$ \vec{u}=\left(\begin{array}{c}x_b-x_a \\ y_b-y_a \end{array}\right)$$ is a direction vector of the line.
then the scalar product of $n$ et $u$ must be $0$ to say that $n$ is indeed a normal vector. $$ \vec{n}\cdot \vec{u}=a(x_b-x_a)+b(y_b-y_a)=ax_b-ax_a+by_b-by_a=c-c=0 $$
Then to show that $\vec{v}=\left(\begin{array}{c}-b \\ a\end{array}\right)$ is a direction vector for the line all we have to do is to calculate the scalar product of $\vec{n}$ and $\vec{v}$. $$\vec{n}\cdot \vec{v}=-ab+ab=0 $$ So the vector $\vec{v}$ is orthogonal to the vector $\vec{n}$ which is a normal vector, hence $\vec{v}$ is a direction vector.
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3$\begingroup$ Good answer. Maybe it is not the best idea to use the same variable name for the coefficients and points on the line. This might confuse. $\endgroup$ Jan 8, 2018 at 8:14
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$\begingroup$ thanks you for the remark, i hope it is less confusing now. $\endgroup$ Jan 8, 2018 at 11:14
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A vector parallel to the line is obtained by joining two points, say
$$\vec p=(x_1,y_1)-(x_0,y_0).$$
Let $\vec n:=(3,4)$. Then by the equation of the line ($3x+4y=1$),
$$\vec n\cdot\vec p=(3x_1+4y_1)-(3x_0+4y_0)=1-1=0$$
which shows that $\vec n\perp\vec p$.
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Two lines are perpendicular if their slopes are opposite reciprocal. The slope of your line is $-3/4$ so you are looking for a line with slope of $4/3$. If you let $x=3t$, $y=4t$ the slope will be 4/3 and you are done.
answered Jan 8, 2018 at 4:38
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$\begingroup$ whats
there? putting3tinplace ofxand4tin place ofydoesnt make sense (3t+4t-1=0is7t-1=0) $\endgroup$– A.BSep 19, 2020 at 19:03
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Short answer:
The line parallel to the given one passing through the origin has equation
$$3x+4y=0,$$ or $$\vec n\cdot\vec p=0.$$
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Completely analogous to the case of a plane. .. $ax+by+cz=d $ has normal vector $(a,b,c) $... this is because the dot product $(x-x_0, y-y_0, z-z_0)\cdot (a,b,c)=0$, where $d=ax_0+by_0+cz_0$, for any point $(x_0, y_0, z_0) $ on the plane. ..
I suspect this generalizes to higher dimensions (somehow). ..
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1$\begingroup$ I assume that if the OP is asking this elementary question about 2D lines, he knows even less about 3D analytical geometry. $\endgroup$– user65203Jan 8, 2018 at 8:24