Question

Let z=xy 2 +xy 3
1a. Find a tangent vector at (2,1,4) that is parallel to the XZ plane (i.e. projection of vector onto XY plane

is parallel to the X axis

b Find an equation to the tangent line at (2,1,4) that is parallel to the XZ plane (i.e. projection of vector onto XY plane is parallel to the X axis

c Find a tangent vector at (2,1,4) that is parallel to the YZ plane (i.e. projection of vector onto XY plane is parallel to the Y axis

d Find an equation to the tangent line at (2,1,4) that is parallel to the XZ plane (i.e. projection of vector onto XY plane is parallel to

Answer 1

@Festinger can u help me

Answer 2

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Answer 3

For the first part, the normal vector will be \(\vec n = \nabla \varphi\) where \(\varphi\) is the surface in level form \(\varphi = xy^2 + xy^3 - z = const\)

so \(\vec n = <y^2 + y^3, 2xy + 3xy^2, -1>\) which you can compute for \((2,1,4)\)

There will be an infinite number of tangent directions vectors \(\vec t = <t_1, t_2, t_3 > \) at the point but we know 2 things.

\(\vec t \cdot \vec n = 0\)

and

\(t_2 = 0\) as \(\vec t\) is parallel with x-z plane

so solve

\(\vec n \cdot <t_1, 0, t_3 > = 0\) at the point \((2,1,4)\) to find out how \(t_1\) and \(t_3\) are related

repeat for other bits

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you can also calc the normal vector for surface \( \vec r = \left[ \begin{matrix}
x \\
y \\
xy^2 + xy^3
\end{matrix} \right ]
\)

as cross product of tangent vectors:

\(\vec n = \dfrac{\partial \vec r }{\partial x} \times \dfrac{\partial \vec r }{\partial y} \) at the point,.....