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Equations of planes ha touc hed on equations of planes previously Here will ﬁll in some of the details. Planes in oin t-normal form The basic data whic determines plane is oin in the plane and ector orthogonal to the plane. call normal to the plane and will sometimes sa is normal to the plane, instead of orthogonal. No w, supp ose an the equation of plane and ha oin in the plane and ector a, b, normal to the plane. Let x, an arbi trary oin in the plane. Then th ector is in the plane and therefore orth ogonal to This means a, b, call this last equation the oin t-normal form for the plane. x, Example 1: Find the pl ane through the oin (1,4, 9) with normal Answ er: oin t-normal form of the plane is 2( 1) 3( 4) 4( 9) 0. can also write this as 50. Example 2: Find the plane con tainin the oin ts (1 3), (0 3), (2 5). Answ er: The goal is to nd the basic data, i.e. oin in the plane and normal to the plane. The oin is easy already ha three of them. get the nor al note (see ﬁgu re elo w) that are ectors in th plane, so their cross pro duct is and orthogonal (normal) to the plane. That is, 2) 1) x,

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Using oin t-normal form (with oin the equation of the lane is 4( 1) 2( 2) 3) or equiv alen tly Example 3: Find the pl ane with normal con taining the oin (0,0,3) Eq. of plane: x, 3. Example 4: Find the pl ane with and in tercepts and Answ er: could ﬁnd this using the metho example 1. Instead, e’ll use shortcut that ork when all the in tercepts are kno wn. In this case, the in tercepts are a, 0) (0 b, 0) (0 and simply write th plane as x/a /b /c ou can easily hec that eac of the giv en oin ts is on the plane. or completeness e’ll do this using the more general metho of example 1. The oin ts giv ectors in the plane, a, b, and a, a, b, a, bc, ac, ab oin t-normal form: bc ac 0) ab 0) bc ac ab abc x/a /b /c 1. Lines in the plane While e’re at it, le t lo ok at ys to write th equation of line in the xy -plane. Slop e-inter ept form Giv en the slop and the -in tercept the equation of lin can written mx Point-normal form can also use oin t-normal form to ﬁnd the equation of line. Giv en oin on the line and ector a, normal to the line the equation of the line can written 0. a,

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IT OpenCourseWare http://ocw.mit.edu 18.0 SC Multivariable Calculus Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

Planes in oin tnormal form The basic data whic determines plane is oin in the plane and ector orthogonal to the plane call normal to the plane and will sometimes sa is normal to the plane instead of orthogonal No w supp ose an the equation of plane ID: 23047

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Page 1

Equations of planes ha touc hed on equations of planes previously Here will ﬁll in some of the details. Planes in oin t-normal form The basic data whic determines plane is oin in the plane and ector orthogonal to the plane. call normal to the plane and will sometimes sa is normal to the plane, instead of orthogonal. No w, supp ose an the equation of plane and ha oin in the plane and ector a, b, normal to the plane. Let x, an arbi trary oin in the plane. Then th ector is in the plane and therefore orth ogonal to This means a, b, call this last equation the oin t-normal form for the plane. x, Example 1: Find the pl ane through the oin (1,4, 9) with normal Answ er: oin t-normal form of the plane is 2( 1) 3( 4) 4( 9) 0. can also write this as 50. Example 2: Find the plane con tainin the oin ts (1 3), (0 3), (2 5). Answ er: The goal is to nd the basic data, i.e. oin in the plane and normal to the plane. The oin is easy already ha three of them. get the nor al note (see ﬁgu re elo w) that are ectors in th plane, so their cross pro duct is and orthogonal (normal) to the plane. That is, 2) 1) x,

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Using oin t-normal form (with oin the equation of the lane is 4( 1) 2( 2) 3) or equiv alen tly Example 3: Find the pl ane with normal con taining the oin (0,0,3) Eq. of plane: x, 3. Example 4: Find the pl ane with and in tercepts and Answ er: could ﬁnd this using the metho example 1. Instead, e’ll use shortcut that ork when all the in tercepts are kno wn. In this case, the in tercepts are a, 0) (0 b, 0) (0 and simply write th plane as x/a /b /c ou can easily hec that eac of the giv en oin ts is on the plane. or completeness e’ll do this using the more general metho of example 1. The oin ts giv ectors in the plane, a, b, and a, a, b, a, bc, ac, ab oin t-normal form: bc ac 0) ab 0) bc ac ab abc x/a /b /c 1. Lines in the plane While e’re at it, le t lo ok at ys to write th equation of line in the xy -plane. Slop e-inter ept form Giv en the slop and the -in tercept the equation of lin can written mx Point-normal form can also use oin t-normal form to ﬁnd the equation of line. Giv en oin on the line and ector a, normal to the line the equation of the line can written 0. a,

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IT OpenCourseWare http://ocw.mit.edu 18.0 SC Multivariable Calculus Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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