# Parametric curve and normal

1. Jun 5, 2010

1. The problem statement, all variables and given/known data

Let $c(t) = ( cos(At), sin(At), 1)$ be a curve. (A is a constant)

Show that the normal to $c(t)$ is always directed toward the z-axis.

3. The attempt at a solution

I am not sure how to show this. (For example, is the question "asking" us to show the cross product of something is 0 ?) If you tell me how to start the problem, I should have no problem.

I have found the normal, which is $N(t) = ( -cos(At), -sin(At), 0)$.

Thanks.

2. Jun 5, 2010

### rock.freak667

So you have N=<-cos(At),-sin(At),0> and this is in the form <x,y,z>.

What is z equal to? What are the consequences of the negative sign in terms of direction?

3. Jun 5, 2010

$z = 1$ ? I mean $z$ is always equal to 1, unless you ask what $z$ is for the normal, which is 0. I'm not sure about your second question, could you explain more? Thanks.

4. Jun 5, 2010

### rock.freak667

I meant for the normal. If z=0, then you're normal is essentially in the xy-plane right?

As for my other question if you have positive values of x and y, in relation to the z-axis, where would you plot those numbers? (Away or toward the axis when you keep increasing positively?)

5. Jun 5, 2010

Away the z axis?

6. Jun 5, 2010

### rock.freak667

Right, so if you have <x,y,0> it points away from the z-axis. Where would <-x,-y,0> point?

7. Jun 5, 2010

Directed toward the axis. I have a question though, cos(At) and sin(At) aren't always positive, so does this still work?

Thanks.

8. Jun 5, 2010

### rock.freak667

I believe if you draw it out, you will see that when cosine is +ve, sine is -ve so one part of the normal will point towards the z-axis and when sine is +ve and cosine is -ve, the other part of the normal points towards the z-axis. In essence it will always point towards the z-axis.

9. Jun 5, 2010