Dot product: normal and tangent

[sci-doo]

Homework Statement

Tangent plane goes through point P=(a,b,f(a,b)). Any point on the plane is then
Q=(x,y,z)=(x,y,f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)) (f_x and f_y are partial derivatives)
and the vector \overline{PQ} is on tangent plane.
Calculate dot product n.\overline{PQ} and show that the normal vector is perpendicular to the tangent plane.

Homework Equations

n=f_x(a,b)\hat{i} + f_y(a,b)\hat{j} - \hat{k}

The Attempt at a Solution

Ok, I should get the dot product = 0.

I don't know how to do that, because I don't know how to get the vector \overline{PQ} into a right form. So far I've tried

\overline{PQ} = \overline{Q} - \overline{P}
= xi + yj +f(x,y)k - ai - bj - f(a,b)k
= (x-a)i + (y-b)j + (f(x,y)-f(a,b))k

How am I supposed to get zero dot product from THAT and n?
n has partial derivatives, can I write them some other way? I'm stuck.

 

[Mark44]

So start with actually doing the dot product of PQ and n. What do you get?

 

[sci-doo]

So start with actually doing the dot product of PQ and n. What do you get?

I get

n dot PQ = f_x(a,b)(x-a)+f_y(a,b)(y-b)-f(x,y)+f(a,b)

I still have those partial derivatives, and I don't know what to do with them.

 

[Mark44]

That's not what I get. You have an error in what you got for PQ.

 

[sci-doo]

That's not what I get. You have an error in what you got for PQ.

To get PQ I think I should start with Q-P. But it is scalar and I don't see how it turns into a vector.

Q-P= x,y,f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) - a,b,f(a,b)

 

[Mark44]

Q is a vector and P is a vector, and Q - P is a vector, and is equal to PQ. When you add or subtract vectors, you do so component-wise.

 

[sci-doo]

Q is a vector and P is a vector, and Q - P is a vector, and is equal to PQ. When you add or subtract vectors, you do so component-wise.

Q and P are given as scalars. P=(a,b,f(a,b)) is, I think, relatively easy to vectorize
ai + bj + f(a,b)k
Q is a bit ugly
xi + yj + f(a,b)k + f_x(a,b)xi - f_x(a,b)ai + f_y(a,b)yj - f_y(a,b)bj
and so Q - P is
xi + yj + f_x(a,b)xi - f_x(a,b)ai + f_y(a,b)yj - f_y(a,b)bj - ai - bj
and as k is gone, can be put as i and j
(x + f_x(a,b)x - f_x(a,b)a -a)i + (y + f_y(a,b)y - f_y(a,b)b -b)j

Taking dot product with n :

(x + f_x(a,b)x - f_x(a,b)a -a)*(f_x(a,b)) + (y + f_y(a,b)y - f_y(a,b)b -b)*(f_y(a,b)) = ..

The product doesn't cancel out, it is not zero.

 

[quantumdude]

I don't know how you're getting that \vec{PQ}, but the dot product with the normal vector is obviously zero.

Q is a bit ugly
xi + yj + f(a,b)k + fx(a,b)xi - fx(a,b)ai + fy(a,b)yj - fy(a,b)bj

?

How do you get terms such as f_x(a,b)x\hat{i}?

f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b) is the z coordinate of Q.

 

[Mark44]

Q and P are given as scalars. P=(a,b,f(a,b)) is, I think, relatively easy to vectorize
ai + bj + f(a,b)k
Q is a bit ugly
xi + yj + f(a,b)k + f_x(a,b)xi - f_x(a,b)ai + f_y(a,b)yj - f_y(a,b)bj

The expression above is incorrect. Your x and y components are correct, but all the rest is the multiplier for the unit vector k.

and so Q - P is
xi + yj + f_x(a,b)xi - f_x(a,b)ai + f_y(a,b)yj - f_y(a,b)bj - ai - bj
and as k is gone, can be put as i and j
(x + f_x(a,b)x - f_x(a,b)a -a)i + (y + f_y(a,b)y - f_y(a,b)b -b)j

Taking dot product with n :

(x + f_x(a,b)x - f_x(a,b)a -a)*(f_x(a,b)) + (y + f_y(a,b)y - f_y(a,b)b -b)*(f_y(a,b)) = ..

The product doesn't cancel out, it is not zero.

 

[sci-doo]

The expression above is incorrect. Your x and y components are correct, but all the rest is the multiplier for the unit vector k.

Thank you for correcting my somewhat unusual (and wrong!) approach..

I didn't see the z-part of Q as it is.

Dot product is (of course):
\bar{PQ}.n = f_x(a,b)(x-a) + f_y(a,b)(y-b) - f_x(a,b)(x-a) - f_y(a,b)(y-b) = 0

Once again, thank you for your help.