Find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, P(x Yo z), and a vectorn = (a, b, c) that is parallel to the line. Then the equation of the line is x - xo = at, y - Yo = bt, z - zo = ct.)x2 - 9xyz + y + 7z = 0, P(1, 1, 1)

Answered: Find parametric equations for the…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.3: Lines

Problem 21E

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Find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, P(x Yo z), and a vector
n = (a, b, c) that is parallel to the line. Then the equation of the line is x - xo = at, y - Yo = bt, z - zo = ct.)
x2 - 9xyz + y + 7z = 0, P(1, 1, 1)

Expert Solution

Step 1

Let $f (x, y, z) = x^{2} - 9 x y z + y^{2} + 7 z^{2}$ . and given point is $P (1, 1, 1)$ .

Partially Differentiating $f$ with respect to $x, y$ and $z$ respectively, we get

$\begin{array}{rcl} \frac{\partial f}{\partial x} & = & \frac{\partial}{\partial x} (x^{2} - 9 x y z + y^{2} + 7 z^{2}) \\ = & 2 x - 9 y z \end{array}$

$\begin{array}{rcl} \frac{\partial f}{\partial y} & = & \frac{\partial}{\partial y} (x^{2} - 9 x y z + y^{2} + 7 z^{2}) \\ = & - 9 x z + 2 y \end{array}$ and

$\begin{array}{rcl} \frac{\partial f}{\partial z} & = & \frac{\partial}{\partial z} (x^{2} - 9 x y z + y^{2} + 7 z^{2}) \\ = & - 9 x z y + 14 z \end{array}$

Therefore at $P (1, 1, 1)$ , that is when $x = 1, y = 1, z = 1$ , we have

$\frac{\partial f}{\partial x} = 2 x - 9 y z = 2 (1) - 9 (1) (1) = - 7 \frac{\partial f}{\partial y} = - 9 x z + 2 y = - 9 (1) (1) + 2 (1) = - 7 \frac{\partial f}{\partial z} = - 9 x y + 14 z = - 9 (1) (1) + 14 (1) = 7$

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