Question

Find the equation of the tangent plane to the given parametric surface at the specified point. If you have software that graphs parametric surfaces, use a computer to graph the surface and the tangent plane

\(x = u + v,y = 3{u^2},z = u - v;\left( {2,3,0} \right)\)

Short answer

The equation of the tangent plane is \(3x - y + 3z = 3\)

Step-by-step solution

Solution for the tangent vector

Let, consider the tangent vector\({r_u}\)and\({r_v}\)

\(\begin{array}{c}{r_u} = \frac{{\partial x}}{{\partial u}}i + \frac{{\partial y}}{{\partial u}}j + \frac{{\partial z}}{{\partial u}}k\\ = i + 6uj + k\\{r_v} = \frac{{\partial x}}{{\partial v}}i + \frac{{\partial y}}{{\partial v}}j + \frac{{\partial z}}{{\partial v}}k\\ = i + (0)j - k\end{array}\)

Solution for the tangent plane

The tangent pane is given by\({r_u} \times {r_v}\)

\(\begin{array}{c}{r_u} \times {r_v} = \left| {\begin{array}{*{20}{c}}i&j&k\\1&{6u}&1\\1&0&{ - 1}\end{array}} \right|\\ = ( - 6u - 0)i - ( - 1 - 1)j + (0 - 6u)k\\ = - 6ui + 2j - 6uk\end{array}\)

Substitute u and v as 1

So, the tangent plane is \( - 6i + 2j - 6k\)

The equation of the tangent plane

Let’s substitute points\(\left( {2,3,0} \right)\)in tangent plane

\(\begin{array}{c} - 6(x - 2) + 2(y - 3) - 6(z - 0) = 0\\ - 6x + 12 + 2y - 6 - 6z = 0\\ - 6x + 2y - 6z + 6 = 0\\3x - y + 3z = 3\end{array}\)

The equation of the tangent plane is \(3x - y + 3z = 3\).