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oh im just simply salivating over this one lol

please take your time, thanks

we take the point; and apply the gradient to it right?

but the initial form seems to be in a vector position equation...

hmm i am trying to find something in my book

(3u) i + (3u^2 +2v) j + (-5v^2) k

the derivative of a vector position is the tangent line .... if i recall correctly

tangent vector that is...

i believe so

3u = 6
3u^2 +2v = -8
-5v^2 = -20
should give us values for u and v maybe?

it's worth a try

so the point is on the tangent plane itself, and not the original equation

yes

then lets find a tangent vector that matches this contraption lol

can we assume u' and v' = 1?

i don't know

im thinking yes ..... and im usually right about this unless im wrong lol

du/du = 1 and dv/dv = 1 is what im thinking...

<3,-6u+2,-10v>
-----------------
<6, -8, -20> right?

you mean add them?

its in the same plane; just have to account for rotation...

-10 means -20; i got no short term memory apparently lol

the approach looks resonable

with 2 vectors on the plane we can cross product them to get the nomal right?

<3,-6u+2,-10v>
-<6, -8, -20>
-----------------
< a, b, c> is an equivalent statement right?

the cross product will give the normal i think

it will; but first we have to determine what the vectors should be :)

<3,-6u+2,-10v>
< r , s, t>
---------------
= 0

watchmath will surely let me know when my stupidity shows tho right ? :)

lets cross these and see what we get... ;)

(-6u+2)(-10v+20) - (8)(-10v)
-[(3.20) - (-6.-10v)]
(3.8) - (-6(-6u+2))

<(60uv -20v -120u +40) , (-60 - 60v) ,(-36u +36)>
maybe ....

(60uv -60v -120u +40)(x-6)
+(-60 - 60v)(y+8)
+(-36u +36)(z+20)
= 0 is what I come up with.....

can we come up with a surface equation from the parametrics used? im sure we can....

x = 3u
y = −3u^2 + 2v
z = −5v^2

iv often wondered if there can be a tangent plane to a curve..... it seems like trying;

i spose if it can be done for a tangent line to a point; its possible for a plane on a curve :)

in order for the coeff of x = 240, then
60uv -2 = 240
60uv = 242
uv = 242/60 if i did it right