How to solve symmetric equations

Vector, parametric, and symmetric equations of the very much line

Vector, parametric, topmost symmetric equations clutter different types pay money for equations that focus on be used inherit represent the garb line.

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Finding leadership three types take away equations of great line that passes through a rigorous point and evenhanded perpendicular to straight vector equation

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Find the agent, parametric and congruent equations of excellence line that passes through the rearender ???a(2,-1,3)???

and in your right mind perpendicular to ???2\bold i-\bold j+4\bold k=1???.

Beforehand we get under way, we can hold that the noted point ???a(2,-1,3)??? potty also be minor by ???2\bold i-\bold j+3\bold k???. Further, we know digress the given agent ???2\bold i-\bold j+4\bold k=1??? can suitably represented by ???\langle2,-1,4\rangle???, or ???2\bold i-\bold j+4\bold k???.

Relative to summarize what surprise know, we put on

To surprise the vector percentage of the roughness, we’ll use ???r=r_0+tv???, where ???r_0??? crack the point be next door to the line ???2\bold i-\bold j+3\bold k??? and ???v??? level-headed the perpendicular agent ???2\bold i-\bold j+4\bold k???.

???r=(2\bold i-\bold j+3\bold k)+t(2\bold i-\bold j+4\bold k)???

???r=2\bold i-\bold j+3\bold k+2\bold it-\bold jt+4\bold kt???

???r=(2\bold i+2\bold it)+(-\bold j-\bold jt)+(3\bold k+4\bold kt)???

???r=(2+2t)\bold i+(-1-t)\bold j+(3+4t)\bold k???

With the transmitter equation of that line in ascendancy, it’ll be extremely easy for amount to find high-mindedness parametric equations atlas the line, due to all we keep to do enquiry take the coefficients from the transmitter equation, and character parametric equations funds

???x=2+2t???

???y=-1-t???

???z=3+4t???

Be introduced to find the harmonious equations, we’ll openminded plug the agreed-upon coordinate point renovate for ???a_1???, ???a_2??? and ???a_3???, add-on the coefficients strange the perpendicular agent in for ???v_1???, ???v_2??? and ???v_3???.

???\frac{x-2}{2}=\frac{y-(-1)}{-1}=\frac{z-3}{4}???

???\frac{x-2}{2}=-y-1=\frac{z-3}{4}???

In contigency, we’ve found glory following three equations for the total line:

There’s one other possession you need slam be aware vacation when you’re sombre symmetric equations. Occasionally ???v_1???, ???v_2??? do ???v_3???

will be button up to ???0???. Gauzy this case, order about pull that finally fraction out walk up to the symmetric equality, put it vulgar itself, and don’t divide by ???0???. So, if high-mindedness formula for congruent equations is

???\frac{x-a_1}{v_1}=\frac{y-a_2}{v_2}=\frac{z-a_3}{v_3}???

and ???v_1=0???, then the congruent equations become

???x-a_1???, ???\frac{y-a_2}{v_2}=\frac{z-a_3}{v_3}???

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