Thread: geometry, parallel vectors...

QUESTION:

Show that the tangent planes of the unit sphere at the points (x,y,0) are parallel to the z axis.

MY WORK:

I get the tangent planes as x^2 + y^2 = 1 ...

in R3 that's the vector (cos(t), sin(t), z) right?

And the z axis is the vector (0,0,z)

so the dot product is z^2

And the product of the magnitudes is sqrt(1 + z^2)*z

cos(theta) = z/sqrt(1+z^2) ??

But I need cos(theta) to be 1 for those vectors to be parallel...

HELP PLEASE! What have I done?

Originally Posted by RanDom

QUESTION:

Show that the tangent planes of the unit sphere at the points (x,y,0) are parallel to the z axis.

MY WORK:

I get the tangent planes as x^2 + y^2 = 1 ...

in R3 that's the vector (cos(t), sin(t), z) right?

And the z axis is the vector (0,0,z)

so the dot product is z^2

And the product of the magnitudes is sqrt(1 + z^2)*z

cos(theta) = z/sqrt(1+z^2) ??

But I need cos(theta) to be 1 for those vectors to be parallel...

HELP PLEASE! What have I done?

x^2 + y^2 = 1 is definitely not the equation of a plane. I've barely glanced at this question but from my brief glance I think you'd be well advised to consider the coordinates of the points as rather than (x,y,0) .......

OK, so...

the unit sphere is



At a point the tangent plane is



Cancel a factor of 2 and use



for the point that's

... and then?

I'm really lost

Originally Posted by RanDom

OK, so...

the unit sphere is



At a point the tangent plane is



Cancel a factor of 2 and use

Mr F adds a small correction in red.

for the point that's

... and then?

[snip]

And then you're nearly done. A normal to the plane is ....

ah, which is perpendicular to (0,0,1), making the plane parallel... thanks