Calculus II Problem: Describe the region

[lizbeth]

Homework Statement

Sorry, this should say "Calculus III problem".

Describe the region that contains all planes passing through (4, 5, 6) that are parallel to <1, 2, 3>.

Homework Equations

The Attempt at a Solution

I cannot imagine how to answer this. I expect that there are multiple planes that could pass through a single point and be parallel to a vector, since the vector can move around in space. But what would that look like? Would the planes form a sphere as you move the vector around the point in a circle? Very confused by this question.

 

[Simon Bridge]

How can a plane be said to be "parallel" to a vector?

 

[LCKurtz]

Homework Statement

Sorry, this should say "Calculus III problem".

Describe the region that contains all planes passing through (4, 5, 6) that are parallel to <1, 2, 3>.

Homework Equations

The Attempt at a Solution

I cannot imagine how to answer this. I expect that there are multiple planes that could pass through a single point and be parallel to a vector, since the vector can move around in space. But what would that look like? Would the planes form a sphere as you move the vector around the point in a circle? Very confused by this question.

I think it is confusing too. Think about the line through (4,5,6) with direction vector <1,2,3>. Now pass a plane through that line. Think of the line as a straight wire and the plane glued to it. Now if you grab that wire and rotate it around its own axis, that plane will swing around sweeping out all of 3 space. Maybe that's what is wanted.

 

[Simon Bridge]

OR - a plane is parallel to a vector if it's normal vector is parallel to it ... not happy with that either because that is just a plane and there is only one of them that goes through that point and "a plane" is not much of a description.
BUT maybe, by "description", then mean the equation of the plane?

 

[lizbeth]

I don't know if this is supposed to be a narrative description or an equation, either. I could not think how to find the equation of the plane given this information. I could find the equation of the line, but not the plane. Wouldn't I need a point outside the plane? Plus, it says "All Planes", not "The Plane". And "planes" is in bold in the homework as well.

 

[Simon Bridge]

Then you need to clear up the ambiguity.
You may have a previous example in your notes - otherwise you'll have to ask the person who set the problem of someone else.

As a final resort, you can provide each interpretation of the question with the answer.

 

[LCKurtz]

OR - a plane is parallel to a vector if it's normal vector is parallel to it ... not happy with that either because that is just a plane and there is only one of them that goes through that point and "a plane" is not much of a description.
BUT maybe, by "description", then mean the equation of the plane?

I don't agree with that at all. A plane is parallel to a vector if the vector is perpendicular to the plane's normal.

 

[Ray Vickson]

Homework Statement

Sorry, this should say "Calculus III problem".

Describe the region that contains all planes passing through (4, 5, 6) that are parallel to <1, 2, 3>.

Homework Equations

The Attempt at a Solution

I cannot imagine how to answer this. I expect that there are multiple planes that could pass through a single point and be parallel to a vector, since the vector can move around in space. But what would that look like? Would the planes form a sphere as you move the vector around the point in a circle? Very confused by this question.

Imagine the vector <1,2,3> sticking out from the point (4,5,6); this will be part of the line $(x,y,z) = (4,5,6) +t(1,2,3) = (4+t,5+t,6+t)$. Now put a plane through this line; rotate the plane about the line and through 360 degrees. That sweeps out the region you are asked for, and it is the entire 3-dimensional space, as LCKurtz has already indicated. (In other words, any point (x,y,z) is on some plane through (4,5,6) that is parallel to <1,2,3>.)

 

[lizbeth]

That is what I was thinking as well, Ray and LCKurtz. Thank you all so much for your help with this.