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Multivariable Calculus

This question is asking me to match the given planes with one or more of these four descriptions: (a) Goes through the origin. (b) Has a normal vector parallel to the xy-plane. (c) Goes through the point (0, 8, 0). (d) Has a normal vector whose dot products with  i ,  j , k are all positive.

The two planes are: 7x + y + 5z − 8 = 0

and

x = 8

I'm assuming that to determine if a plane goes through the origin you plug in the points (0,0,0) and see if it satisfies the equation. This also goes for determining if the plane goes through the point (0, 8, 0) you plug it in the equation and see if it satisfies it.

I am unsure how to determine if the descriptions (a) and (d) satisfy the plane. Could someone explain how I could figure that out?

What I got for my answers: 7x + y + 5z − 8 = 0 - (a) (c) x = 8 - I am unsure about this one but I am assuming this plane does not go through the origin and does not go through the point (0,8,0).

Thanks for your help.

2 Answers 2

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(a) If a plane goes through the origin, then the point $(0,0,0)$ will satisfy the equation of that plane.

Plane 1: $7x + y + 5z − 8 = 0$

$LHS=7(0)+(0)+5(0)-8=-8\neq RHS$ : This plane does not go through the origin.

Plane 2: $x = 8$

$LHS=0 \neq RHS$ : This plane does not go through the origin.

(b) The normal vector of a plane with equation $ax+by+cz+dx=0$ is given by $n=ai+bj+ck$. A vector parallel to the $xy$ plane would be perpendicular to the normal to the $xy$ plane, so we need $(ai+bj+ck).(k)=0$

This is equivalent to $c=0$.

Plane 1: $7x + y + 5z − 8 = 0$

$c=5$ : This plane does not have a normal vector parallel to the $xy$ plane.

Plane 2: $x = 8$

$c=0$ as required: This plane does have a normal vector parallel to the $xy$ plane.

(c) is similar to (a)

(d) is similar to (b)

1

your generic plane

$ax +by + cz = d$

Goes through the origin iff:

$a0 + b0 + c0 = d\\ d = 0$

Has a normal vector parallel to the xy-plane.

$ax +by + cz = d$ has normal vector $a\mathbf i +b \mathbf j +c \mathbf k$ the xy plane $= m\mathbf i +n \mathbf j$

What does that say about $c?$

Goes through the point $(0,8,0):$

$8b = d$

Has a normal vector whose dot products with  $\mathbf i ,  \mathbf j , \mathbf k$ are all positive.

$(a\mathbf i +b \mathbf j +c \mathbf k)\cdot (\mathbf i) = a > 0$ continue for $b,c$