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Question Number 80943 by Kunal12588 last updated on 08/Feb/20

Find equation of a plane passing through the points (x_1 , y_1 , z_1 ), (x_2 , y_2 , z_2 ) and perpendicular to the plane ax+by+cz=d

Findequationofaplanepassingthroughthepoints(x1,y1,z1),(x2,y2,z2)andperpendiculartotheplaneax+by+cz=d

Answered by mr W last updated on 08/Feb/20

say projection of P(x_1 ,y_1 ,z_1 ) an the plane ax+by+cz=d is P ′(x_3 ,y_3 ,z_3 ) x_3 =x_1 +at y_3 =y_1 +bt z_3 =z_1 +ct a(x_1 +at)+b(y_1 +bt)+c(z_1 +ct)=d ax_1 +by_1 +cz_1 +(a^2 +b^2 +c^2 )t=d ⇒t=((d−ax_1 −by_1 −cz_1 )/(a^2 +b^2 +c^2 )) ⇒x_3 =x_1 +((a(d−ax_1 −by_1 −cz_1 ))/(a^2 +b^2 +c^2 )) ⇒y_3 =y_1 +((b(d−ax_1 −by_1 −cz_1 ))/(a^2 +b^2 +c^2 )) ⇒z_3 =z_1 +((c(d−ax_1 −by_1 −cz_1 ))/(a^2 +b^2 +c^2 )) the requested plane passes through (x_1 ,y_1 ,z_1 ),(x_2 ,y_2 ,z_2 ),(x_3 ,y_3 ,z_3 ) and its equation is determinant (((x−x_1 ),(y−y_1 ),(z−z_1 )),((x_2 −x_1 ),(y_2 −y_1 ),(z_2 −z_1 )),((x_3 −x_1 ),(y_3 −y_1 ),(z_3 −z_1 )))=0 or determinant (((x−x_1 ),(y−y_1 ),(z−z_1 )),((x_2 −x_1 ),(y_2 −y_1 ),(z_2 −z_1 )),(((a(d−ax_1 −by_1 −cz_1 ))/(a^2 +b^2 +c^2 )),((b(d−ax_1 −by_1 −cz_1 ))/(a^2 +b^2 +c^2 )),((c(d−ax_1 −by_1 −cz_1 ))/(a^2 +b^2 +c^2 ))))=0

sayprojectionofP(x1,y1,z1)antheplaneax+by+cz=disP′(x3,y3,z3)x3=x1+aty3=y1+btz3=z1+cta(x1+at)+b(y1+bt)+c(z1+ct)=dax1+by1+cz1+(a2+b2+c2)t=d⇒t=d−ax1−by1−cz1a2+b2+c2⇒x3=x1+a(d−ax1−by1−cz1)a2+b2+c2⇒y3=y1+b(d−ax1−by1−cz1)a2+b2+c2⇒z3=z1+c(d−ax1−by1−cz1)a2+b2+c2therequestedplanepassesthrough(x1,y1,z1),(x2,y2,z2),(x3,y3,z3)anditsequationis|x−x1y−y1z−z1x2−x1y2−y1z2−z1x3−x1y3−y1z3−z1|=0or|x−x1y−y1z−z1x2−x1y2−y1z2−z1a(d−ax1−by1−cz1)a2+b2+c2b(d−ax1−by1−cz1)a2+b2+c2c(d−ax1−by1−cz1)a2+b2+c2|=0

Commented by peter frank last updated on 08/Feb/20

sir if you get time please help Qn 80199

sirifyougettimepleasehelpQn80199

Commented by Kunal12588 last updated on 08/Feb/20

thank you very much sir. while I was doing :) suppose eq^n of req. plane is a_1 x+b_1 y+c_1 z=d_1 given points satisfy the plane ∴ (i) a_1 x_1 +b_1 y_1 +c_1 z_1 =d_1 (ii) a_1 x_2 +b_1 y_2 +c_1 z_2 =d_1 req. plane ⊥ (ax+by+cz) ⇒ (iii) a_1 a+b_1 b+c_1 c=0 I only knew this much 3 eq^n s 4 unknowns(a_1 ,b_1 ,c_1 ,d_1 ). what you did is find a point in the line of intersection of the planes. okay I will remember. Thanks once again sir.

thankyouverymuchsir.whileIwasdoing:)supposeeqnofreq.planeisa1x+b1y+c1z=d1givenpointssatisfytheplane∴(i)a1x1+b1y1+c1z1=d1(ii)a1x2+b1y2+c1z2=d1req.plane⊥(ax+by+cz)⇒(iii)a1a+b1b+c1c=0Ionlyknewthismuch3eqns4unknowns(a1,b1,c1,d1).whatyoudidisfindapointinthelineofintersectionoftheplanes.okayIwillremember.Thanksonceagainsir.

Commented by mr W last updated on 08/Feb/20

there are many ways to get the eqn.

therearemanywaystogettheeqn.

Commented by mr W last updated on 08/Feb/20

for the eqn. of the requested plane a_1 x+b_1 y+c_1 z=d_1 you actually only need to know the ratios of the coefficients, e.g. a_1 x+b_1 y+c_1 z=1 then you have three unknowns and three equations.

fortheeqn.oftherequestedplanea1x+b1y+c1z=d1youactuallyonlyneedtoknowtheratiosofthecoefficients,e.g.a1x+b1y+c1z=1thenyouhavethreeunknownsandthreeequations.

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