231.12.3.63 Determine the smallest distance between point and line

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SUMMARY

The discussion focuses on calculating the smallest distance between a point \( P(1,1,1) \) and a line \( L \) defined by the direction vector \( \langle -4,-5,8 \rangle \) originating from the origin. The formula used for this calculation is \( d = \frac{|Am + Bn + C|}{\sqrt{A^2 + B^2}} \). Participants clarify the need to derive the equation for the line \( L \) to apply the distance formula correctly. The conversation also references a similar question, indicating the relevance of understanding the geometric relationship between points and lines in three-dimensional space.

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$\tiny{231.12.3.63}$
$\textsf{Determine the smallest distance between point $P(1,1,1)$}$
$\textsf{ and the line $L$ through the origin}$
$\textsf{ $L$ has the direction $\langle -4,-5,8 \rangle$}$

$\textit{ok I know the formula to use basically is}$

\begin{align*}\displaystyle
d&=\frac{|Am+Bn+C|}{\sqrt{A^2+B^2}}
\end{align*}

$\textit{But did not know how to get the equation for L}$
 
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Isn't this question a duplicate of the following?

http://mathhelpboards.com/calculus-10/231-12-3-65-determine-smallest-distance-between-point-line-22060.html
 

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