The accompanying figure illustrates this problem:[br][br][b]Suppose two surfaces intersect to form a curve, [math]C[/math], and suppose [math]P_0[/math] is a point on [math]C[/math]. How do you find a parametric equation for the line tangent to [math]C[/math] at [math]P_0[/math]?[/b][br][br]To solve the problem, we observe that the tangent line is orthogonal to both [math]\nabla f[/math] and [math]\nabla g[/math] at [math]P_0[/math], and therefore parallel to [math]\mathbf{v}=\nabla f\times\nabla g[/math]. The components of [math]\mathbf{v}[/math] and the coordinates of [math]P_0[/math] give us equations for the line.

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]