planes {cal3}

Planes:

• Vector eq of plane?
• Given some vec $\vec{n}=<a,b,c>,\vec{P}=<x_0,y_0,z_0$ then nP
• Scalar EQ of plane:
  • a(x-b)
• What about EQ of planes in other cord sys? eg.
  • Parametric form: As many vars as obj has dimensions (line has 1 dimension → js t) (plane is 2d, then use s&r)
    • Eg. $r(s,t)=<X(s,t),Y(s,t),Z(s,t)>$

Planes: (copy form lect board)

1. EQ form: $ax+by+cz=d$
2. Func form? f(x,y) = ax+by+c / z=ax+by+c
3. $r(s,t)=<X(s,t),Y(s,t),Z(s,t)>$

Just know abt EQ form + putting them into func form
eg.1) Create plane parallel to a = <1,1,0>. b = <1,0,1>.

1. Take 2 tan vectors turn into normal vect? Take cross product (since 3d)
   1. a x b → c, where c should be
2. Take Normal vect + point and convert into EQ form

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• Convert 2 pts (n and p) to a plane
• Convert 3 pts to a plane?
  • (See lamr edu & notes)

Distance pt 2 pt

• in 2D: solve for y
• Make a new vect func L(t)=<t,f(x)>
• Find the derative of L(t) to get tan vect ()

Notes

• Everything regarding vect can be setup into a sys of EQ
• Know: 3 pts to EQ of plane. projecting a unto b,

Quiz Stuff

• Given 3 vectors:
  • Make a plane with 2 vectors tangent to plane and one vector as point
  • Make plane with a vector and a normal vector