Question

for each situation write a vector equation and a scalar equation of the plane.
a) perpendicular to the line [x,y,z]=[2,4,9]+t[3,5,-3]
b)parallel to the yz-plane and including the point (-1,-2,5)
c)parallel to the plane 3x-9y+z-12=0 and including the point (-3,7,1)

Answer 1

d.m=0

Answer 2

so [x,y,z].[3,5,-3]=0

Answer 3

so 3x+5y-3z=0

Answer 4

right, you got it quicker than me :)

Answer 5

hahaha you taught me :P

Answer 6

:)
I'm getting tired I guess
ok part b)

Answer 7

but for a what do we do for 3x+5y-3z=0

Answer 8

?

Answer 9

if it's parallel to the yz-plane then the normal vector should be \(\vec i\) in the x-direction, eh?\[\vec n=\langle 1,0,0\rangle\]

Answer 10

i dont get it they are asking of vector and scalar equation including the point (4,-2,7)

Answer 11

you say (-1,-2,5) in your post

Answer 12

thsts for b

Answer 13

but a perpendicular (4,-2,7)

Answer 14

the equation of some plane is derivable from some point P=<x,y,z> and a specific point \(P_0=<a,b,c>\)\[\vec n\cdot\vec P=\vec n\cdot\vec P_0\implies\vec n\cdot(\vec P-P_0)=\vec n\cdot(\langle x,y,z\rangle-\langle a,b,c\rangle)=0\]\[\vec n\cdot\langle x-a,y-b,z-c\rangle=0\]

Answer 15

in your case, \(P_0=\langle4,-2,7\rangle\) and \(\vec n=\langle 1,0,0\rangle\) for part b)
and \(P_0=\langle-3,7,1\rangle\) and \(\vec n=\nabla f(x,y,z)\) for part c)