I have the right to see that from the first plane I obtain the vector $\\langle5,1,-1\\rangle$ indigenous the coefficients and also then indigenous the 2nd plane I obtain coefficients that $\\langle -25,-5,5\\rangle$.

Straight far I have the right to see that vector $2$ is just vector $1$ multiplied by $-5$, but and I would have thought this was sufficient to suggest parallel, yet I doubt myself since of the answers at the en perform the equations.

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If i multiplied every of plane $1$ through $-5$ I\"d acquire $-25x -5y + 5z = -35$. That is the $-35$ and also the $9$ that confuse me, if the planes room equal must these two numbers likewise be multiples of $-5$ ?


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Recall that planes room parallel in $3$ size if they do not share any points. Since every allude $P=(x,y,z)$ top top the very first plane satisfies $$5x+y-z=7$$

we recognize (as girlfriend pointed out) that every suggest on the an initial plane likewise satisfies

$$-25x-5y+5z=-35$$

However, due to the fact that $-35\\ne 9$ no point $P$ ~ above the an initial plane can satisfy the equation that the second plane, for this reason the 2 planes carry out not share any points and are thus parallel.

However if the 2nd equation had $-35$ rather of $9$ ~ above the right-hand side, then the 2 planes would undoubtedly be the very same (since one equation is satisfied specifically when the various other is).


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Peter WoolfittPeter Woolfitt
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That is adequate to show that the planes space parallel. I\"m no sure how much friend know about vectors yet the means to do it with vectors is:Let $\\thedesigningfairy.combf n = (5,1,-1),\\;\\thedesigningfairy.combf m = (-25,5,5) = -5\\thedesigningfairy.combf n$ climate the equations of your planes are:\\beginalign\\Pi_1:&&\\thedesigningfairy.combf r\\cdot\\thedesigningfairy.combf n &=7\\\\\\Pi_2:&&\\thedesigningfairy.combf r\\cdot\\thedesigningfairy.combf m &=9\\\\\\endalignwhich is indistinguishable to\\beginalign\\Pi_1:&&\\thedesigningfairy.combf r\\cdot\\thedesigningfairy.combf n &=7\\\\\\Pi_2:&&\\thedesigningfairy.combf r\\cdot\\thedesigningfairy.combf n &=-\\frac 95\\\\\\endalignbecause $m=-5n.$ clearly at most one the those equations is to solve for any given $\\thedesigningfairy.combf r$ for this reason the planes should be parallel together they cannot intersect.


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Dan RobertsonDan Robertson
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