Find the value of $\displaystyle a$ such that the planes $\displaystyle ax+y+z=0$, $\displaystyle x+3z=0$, and $\displaystyle 5y+6z=0$ have a line in common.

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- Jan 23rd 2009, 11:41 PMalexmahone3D geometry
Find the value of $\displaystyle a$ such that the planes $\displaystyle ax+y+z=0$, $\displaystyle x+3z=0$, and $\displaystyle 5y+6z=0$ have a line in common.

- Jan 24th 2009, 01:02 AMearboth
Let

$\displaystyle p_1:x+3z=0$

$\displaystyle p_2:5y+6z=0$ and

$\displaystyle p_3:ax+y+z=0$

denote the three planes.

1. Calculate

$\displaystyle p_1 \cap p_2 = l_{p_1,p_2} : \left\{\begin{array}{l}x=-3t \\y=-\frac65t \\z = t\end{array}\right.$

2. Calculate

$\displaystyle p_2 \cap p_3 = l_{p_2,p_3} : \left\{\begin{array}{l}x=\frac1{5a} t \\y=-\frac65t \\z = t\end{array}\right.$

3. Compare the coefficients:

$\displaystyle l_{p_1,p_2} = l_{p_2,p_3}\ if\ -3=\frac1{5a}~\implies~ a=\dfrac1{15}$

4. Thus the third plane has the equation:

$\displaystyle p_3:\dfrac1{15} x+y+z=0$

5. I've attached a sketch of the three planes. - Jan 24th 2009, 01:43 AMalexmahone
- Jan 24th 2009, 03:39 AMmr fantastic
- Jan 24th 2009, 04:12 AMearboth