Unit vector tangent to the surface

[katielouise]

I have the following question:

Given that ø = (x^2)y + cos(z) find the unit vector n which is both tangent to the surface of constant ø at (1,1,∏/2) and normal to the vector b = x + y - 2z (where x y and z are the unit vectors)

I have calculated ∇ø = 2x + y - z (again where x y and z are the unit vectors)

but I am unsure what to do next.

If I want the unit vector tangent to this surface then its gradient has to be the same as what I calculated above? And if its normal to b then n dotted with b has to = 0? Just don't know how to use this information or anything else to actually calculate the unit vector n.

Thanks :)

 

[Dick]

I have the following question:

Given that ø = (x^2)y + cos(z) find the unit vector n which is both tangent to the surface of constant ø at (1,1,∏/2) and normal to the vector b = x + y - 2z (where x y and z are the unit vectors)

I have calculated ∇ø = 2x + y - z (again where x y and z are the unit vectors)

but I am unsure what to do next.

If I want the unit vector tangent to this surface then its gradient has to be the same as what I calculated above? And if its normal to b then n dotted with b has to = 0? Just don't know how to use this information or anything else to actually calculate the unit vector n.

Thanks :)

If you call the vector you are looking for $a=a_x \hat x + a_y \hat y + a_z \hat z$, then $a \cdot b=0$ and $a \cdot n=0$, write down those equations and see what they tell you about the components of a.

 

[D H]

but I am unsure what to do next.

You could follow Dick's advice. Alternatively, you could use some other operation on a pair of vectors that yields a vector that is normal to both.

 

[katielouise]

You could follow Dick's advice. Alternatively, you could use some other operation on a pair of vectors that yields a vector that is normal to both.

Sorry, I'm still a bit confused. I want a vector that is normal to b but tangent to ∇ø, not normal to both?

 

[D H]

You don't want a vector tangent to ∇ø. The gradient ∇ø points in the direction along which ø(x,y,z) changes the fastest. That is not along a tangent to the surface of constant ø. The gradient normal to this level surface.

 

[katielouise]

You don't want a vector tangent to ∇ø. The gradient ∇ø points in the direction along which ø(x,y,z) changes the fastest. That is not along a tangent to the surface of constant ø. The gradient normal to this level surface.

oh yeah of course! So i do the cross product of ∇ø and b? The vector I get as a result of that, will I then have to divide it by its magnitude to make it into the unit vector?

 

[D H]

Yes. You'll get the same vector (possibly multiplied by -1) with Dick's method.

 

[katielouise]

thanks for your help :)