Point distance from straight

Dec 2018
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Finland
[Solved] Point distance from straight

Hi,
I have done a few tasks, that are the same type, for example: "Calculate the point a = [1, 2] distance from y = 3x + 2. These are easy and simple to do for me, but now I came across to a same subject but a different kind, that I don't really understand

The question is:
"Calculate the point (2, 6) distance from straight r = r0 + tv, when r0 = [1, -1] and v = [2, 10]

I don't understand how to start doing it. Hope for some help.
Thank you.
 
Last edited:
Nov 2013
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2,962
California
$r =(1,-1)+t(2,10)$

can be written

$y+1 = 5(x-1)$

$y = 5x-6$

can you finish?
 
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Dec 2013
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Colombia
$r =(1,-1)+t(2,10)$ can be written
$y+1 = 5(x-1)$
$y = 5x-6$
To clarify: $(1,-1)$ is a point $(x_0,y_0)$ on the line and $(2,10)$ is the direction or slope of the line - how $x$ and $y$ change together.

Thus the gradient $m=\frac{\Delta y}{\Delta x}=\frac{10}2=5$ and the the equation of the line is \begin{align}y-y_0 &= m(x-x_0) \\ y+1 &= 5(x-1)\end{align} as per romsek.
 
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As an alternative approach, you can look for the vector perpendicular to the direction of the line (i.e. $\langle2,10\rangle$) that passes through the point $P=(2,6)$.

The points on the line are $Q_t=\langle1,-1\rangle + t\langle 2,10\rangle$. We just need to find the one for which the vector $\vec{PQ_t}$ joining it to $P$ is perpendicular to $\langle2,10\rangle$.

$$\vec{PQ_t}=\vec{OQ_t} - \vec{OP} = \langle1,-1\rangle + t\langle 2,10\rangle - \langle 2,6 \rangle = \langle 2t-1, 10t-7 \rangle$$

For this to be perpendicular to $\langle2,10\rangle$, we require that the scalar (dot) product of $\langle2,10\rangle$ and $\vec{PQ_t}$ be equal to zero.

That is \begin{align}\vec{PQ_t} \cdot \langle2,10\rangle = \langle 2t-1, 10t-7 \rangle \cdot \langle2,10\rangle &= 0 \\ 2(2t-1) + 10(10t-7) &= 0 \\ 104t - 72 &= 0 \\ t &= \tfrac9{13} \end{align}

Thus the vector $\vec{PQ_t}=\langle 2t-1, 10t-7 \rangle$ with $t=\frac9{13}$ is the perpendicular that joins the line and the point $P$. It's length is the perpendicular distance between the line and $P$.
 
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