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- Apr 29th 2012, 01:43 AMalakdequation of line and plane !! plzzzz help
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- Apr 29th 2012, 02:47 AMProve ItRe: equation of line and plane !! plzzzz help
The line will be infinitely long and in the direction of the vector that goes from $\displaystyle \displaystyle \begin{align*} \overline{OP} \end{align*}$ to $\displaystyle \displaystyle \begin{align*} \overline{OQ} \end{align*}$. So

$\displaystyle \displaystyle \begin{align*} \overline{PQ} &= \overline{PO} + \overline{OQ} \\ &= -\overline{OP} + \overline{OQ} \\ &= \overline{OQ} - \overline{OP} \\ &= <1, 0, 4> - <1, 1, 2> \\ &= <0, -1, 2> \end{align*}$

To make it infinitely long, we multiply by some parameter, $\displaystyle \displaystyle \begin{align*} t \end{align*}$, to give $\displaystyle \displaystyle \begin{align*} t<0, -1, 2> \end{align*}$.

Then we need to position this infinitely long vector at a point. Adding either of the points you know the line goes through will do.

So $\displaystyle \displaystyle \begin{align*} l = <1, 1, 2> + t<0, -1, 2> \end{align*}$.

This is the vector form of the line.

To get the parametric equations, expand it out...

$\displaystyle \displaystyle \begin{align*} l &= <1, 1 - t, 2 + 2t> \\ \\ x &= 1 \\ \\ y &= 1 - t \\ \\ z &= 2 + 2t \end{align*}$

To get the Cartesian equations, try to write each of them as t in terms of the other variable and set them equal.