# Equation of a plane given point and line

Find the equation of the plane containing the point $(1, 3,−2)$ and the line $x = 3 + t$, $y = −2 + 4t$, $z = 1 − 2t$.

Don't know where to start?

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Hint:By taking $t=1$ which can be any other real number into the equation of the line we have: $$P:(4,2,-1)$$ as a point lying on the line. Now write the equation of the line passing $P$ and the point you have above $A:(1,3,-2)$. Then find the cross product of two leading vector, one is for the first line, and the other is for the line $PA$. Use it for the normal vector for the desired plane.

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Nice hint ${\checkmark^{\checkmark}}^{\large +1}$ – amWhy Apr 10 '13 at 0:37

First, lets find the point lying on the line when $t=0$:

$\begin{cases} x = 3 + 0 \\ y = −2 + 0 \\ z = 1 − 0 \end{cases}$ Thus we have $\rightarrow$ $A(3,-2,1)$

Then, find another point, we denote as B, when $t = 1$:

$\begin{cases} x = 3 + 1 \\ y = −2 + 4 \\ z = 1 − 2 \end{cases}$ Thus we have $\rightarrow$ $B(4,2,-1)$

Then determine the vectors on the plane, then obtain the normal vector by taking the cross product which will finally give you the desired equation. If you need more help just ask.

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+1 for a Nice explanation. I hope, she can find everything here via these two answers. – Babak S. Dec 8 '12 at 9:40
@BabakSorouh Thanks Babak, I knew what to do because I read your answer and it reminded me of what to do : ) – diimension Dec 8 '12 at 9:41
Thanks so much guys!! – Linda Dec 8 '12 at 10:11