Tangent Plane
$$\vec n = \vec T_{1} \cdot \vec T_{2} = \left( \begin{matrix} i & j & k \ 0 & 1 & f_{2}(a,b) \ 1 & 0 & f_{1}(a,b) \end{matrix} \right) = f_{1}(a,b)i + f_{2}(a,b)j - k $$
Find a normal vector and equation of the tangent plane and normal line to the graph. $$z = f(x,y) = \sin (xy) \s \text{at} , P\left(\frac{\pi}{3}, -1\right)$$ Calculate $z$ $$z = \sin\left(\frac{-\pi}{3}\right) = \frac{-1}{2}$$ Partial derivatives $$f_{1} = \frac{\partial z}{\partial x} = \frac{\partial}{{\partial x}} \sin(xy) = \cos(xy) \cdot y \cdot 1$$ $$f_{2} = \frac{\partial z}{\partial y} = \frac{\partial}{{\partial y}} \sin(xy) = \cos(xy) \cdot x \cdot 1$$ Calculate $\vec n$ $$ >\vec n = f_{1}(a,b)i + f_{2}(a,b)j - k = \cos\left(\frac{\pi}{3} \cdot (-1)\right) \cdot (-1) + \cos\left(\frac{\pi}{3} \cdot (-1)\right) \cdot \frac{\pi}{3} - k >$$
$$ >\vec n = -\frac{1}{2}i + \frac{1}{2} \cdot \frac{\pi}{3}j - k = - \frac{1}{2}i + \frac{\pi}{6}j - k >$$