Vector equation of a plane containing this line.

[manmachine]

Homework Statement

What is the vector equation of a plane that contains the line y=(3,2,1)+x(1,0,2) and is parallel to the z-axis?

Homework Equations

Equation of a plane: Ax + By + Cz + D = 0

The Attempt at a Solution

Ok so i substitute the values into the plane,
A(1)+B(0)+C(2)+D=0

But then I am left with 3 variables. Can you please offer some help?

Kind Regards,

 

[tiny-tim]

welcome to pf!

hi manmachine! welcome to pf!

sorry, but i don't understand what you're doing at all

the easiest way of finding the equation is probably first to find the normal to the plane …

what will that be? ​

(and i think the question probably means the parametric equation v = (3,2,1) + t(1,0,2))

 

[manmachine]

hi manmachine! welcome to pf!

sorry, but i don't understand what you're doing at all

the easiest way of finding the equation is probably first to find the normal to the plane …

what will that be? ​

(and i think the question probably means the parametric equation v = (3,2,1) + t(1,0,2))

Yes but I don't know what the plane is. I know that the normal is the coefficients if the plane. Please help.

 

[HallsofIvy]

Homework Statement

What is the vector equation of a plane that contains the line y=(3,2,1)+x(1,0,2) and is parallel to the z-axis?

Homework Equations

Equation of a plane: Ax + By + Cz + D = 0

The Attempt at a Solution

Ok so i substitute the values into the plane,
A(1)+B(0)+C(2)+D=0

Why do you substitute those values? Those would be the components of a vector in the plane, not the coordinates of any point in the plane. (3, 2, 1) is a point in the plane. Setting x= 3, y= 2, z= 1, we have 3A+2B+ C+ D= 0.

Also, it is not a good idea to use "x" and "y" in that equation for the plane. In that vector equation "x" is a parameter, not the x- coordinate and "y" is a vector. Better to use, say, r= (3, 2, 1)+ t(1, 0, 2).

But then I am left with 3 variables. Can you please offer some help?

Kind Regards,

In that form, you can always divide through by any non=zero coefficient so you really have only two variables left. To eliminate one, use the fact that the plane is parallel to the z-axis. That means that the normal to the plane is also normal to the z-axis. That is, <0, 0, 1>.<A, B, C>= C= 0.

 

[manmachine]

And yes, but how can I find the normal if don't have to vector equations IAXBI?

 

[tiny-tim]

hi manmachine!

Yes but I don't know what the plane is. .

yes, but you do know the directions of two lines that the normal is perpendicular to, don't you?

 

[manmachine]

There is only one line Tim! I can't use cross product :(

 

[tiny-tim]

z-axis?

 

[manmachine]

oh fudge :P (bangs head on wall)
So it is 0,1,0 by cross product right?

 

[tiny-tim]

oh fudge :P (bangs head on wall)

he he!
​

So it is 0,1,0 by cross product right?

uhh?

 

[manmachine]

Isn't that the normal to plane?

 

[tiny-tim]

(0,1,0) is the y-axis, and is normal to the xz-plane

 

[manmachine]

If I take the cross product of the z-axis (being 0,0,1) and our vector (1,0,2) this is what i get.

 

[tiny-tim]

oh i see!

ok, now you know the normal to the plane, and you know at least one point on the plane,

so the vector equation of the plane is … ? ​

 

[manmachine]

Am i right :)?

Take coefficents
0a+1b+0c+D=0
(0)(3)+(1)(1)+D=0
d=-1

This equation of our is plane is this:
b+-1=0 ?

 

[tiny-tim]

Take coefficents
0a+1b+0c+D=0
(0)(3)+(1)(1)+D=0
d=-1

shouldn't there be a 2 in there somewhere?

This equation of our is plane is this:
b+-1=0 ?

what happened to x y and z ?

don't make it so complicated …

you know it's vertical, and parallel to the xz-plane, so it must be y = constant

ok, what is obviously that constant?? ​

 

[manmachine]

I have no idea :p

 

[manmachine]

What do I do next . I am sorry I just don't get it

 

[HallsofIvy]

This is very peculiar. You say that you know that any plane can be written as Ax+ By+ Cz+ D= 0. It has already been pointed out that you an divide through by any number, say, D, to reduce to three unknown constants, A'x+ B'y+ C'z+ 1= 0 (with A'
= A/D etc.). If the line given by the vector formula r= (3,2,1)+t(1,0,2) or parametric equations x= 3+ t, y= 2, z= 1+ 2t. In particular, taking t= 0, (3, 2, 1) is a point in the plane so we must have 3A'+ 2B'+ C'+ 1= 0.

Also, taking t= 1, (3+ 1, 2+ 0, 1+ 2)= (4, 2, 3) is also a point in the plane. so we must have 4A'+ 2B'+ 3C'+ 1= 0.

The last condition is that the plane is parallel to the z-axis. That means that its normal vector, <A, B, C> or <A', B', C'> is normal to <0, 0, 1>. Their dot product gives C= C'= 0 and so we have the two equations -3A'- 2B'- 1= 0 and 4A'+ 2B'+ 1= 0.

By the way, I find it simpler to use the fact that if a plane has normal vector <A, B, C> and contains point (x_0, y_0, z_0), then the plane has equation A(x- x_0)+ B(y- y_0)+ C(z- z_0)= 0.

 

[tiny-tim]

you know that the plane is y = constant

you also know that (3,2,1) lies on the plane

soooo, y must be … ? ​

 

[manmachine]

3a'+2b'+c'+1=0?