10.3 Example

From [1, p. 440, 36(b)].

Asked: The plane through point $P_0$, (2,-3,2), and the line $6x+4y+3z+5=0$, $2x+y+z-2=0$.

$$\epsffile{vecgeom2.eps}$$

10.3.1 Identification

$$\epsffile{vecgeom2.eps}$$

• I need a vector normal to the plane.
• I can get this by crossing two vectors in the plane.
• One such vector is $\vec n_1 \times \vec n_2$.
• To find another, find any point $Q$ on the line, then $r_Q- r_{P_0}$ is in the plane.

10.3.2 Solution

$$6x+4y+3z+5=0 \quad\quad\Rightarrow\quad\quad \vec n_1 = (6,4,3)$$

$$2x+y+z-2=0 \quad\quad\Rightarrow\quad\quad \vec n_2 = (2,1,1)$$

$$\vec s = \left\vert \begin{array}{ccc} {\hat\imath}& {... ... = \left( \begin{array}{c} 1 0 -2 \end{array} \right)$$

When $x=0$ on the line,

$$4y+3z+5=0, \quad y+z-2=0 \quad\quad\Rightarrow\quad\quad x=0, \quad y=-11,\quad z = 13$$

$$\vec n = \left(\begin{array}{c} 1 0 -2 \end{array} \... ...ft( \begin{array}{c} 2 -3 2 \end{array} \right) \right]$$

$$= \left(\begin{array}{c} 1 0 -2 \end{array} \right) ... ...= \left( \begin{array}{c} -16 -7 -8 \end{array} \right)$$

$$\left(\begin{array}{c} 16 7 8 \end{array} \right) \c... ...ot \left( \begin{array}{c} 2 -3 2 \end{array} \right)$$

$$16 x + 7 y + 8 z = 27$$