Equation of Tangent Plane to Surface

Theorem

Let $\SS$ be a surface embedded in a Cartesian $3$-space.

Let $\TT$ be a tangent plane to $\SS$ at the point $P = \tuple {x_0, y_0, z_0}$.

Then $\TT$ can be expressed using the equation:

$l \paren {x - x_0} + m \paren {y - y_0} + n \paren {z - z_0} = 0$

where $l$, $m$ and $n$ are the direction cosines of the normal to $S$ at $P$.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): tangent plane
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): tangent plane