Is the normal of c(t) always directed toward the z-axis?

[madachi]

Homework Statement

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.

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.

 

[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?

 

[madachi]

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?

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.

 

[rock.freak667]

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.

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?)

 

[madachi]

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?)

Away the z axis?

 

[rock.freak667]

Away the z axis?

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

 

[madachi]

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

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

Thanks.

 

[rock.freak667]

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

Thanks.

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.

 

[madachi]

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.

Thanks. How should we justify the answer though? I am not sure "what to say" to answer the question. Thanks.

 

[rock.freak667]

Thanks. How should we justify the answer though? I am not sure "what to say" to answer the question. Thanks.

Your normal is <-cos(At),-sin(At),0> or x= - cos(At), y= -sin(At), if you sketch this in the xy-plane you will get a circle. Each diameter will be a normal. As long as each one passes through the origin (where the z-axis would be perpendicular to the point (0,0)) that would illustrate it.

The illustration would work I guess.