Rog_Sec_12_5

Rog_Sec_12_5 - Math 20C Lecture Examples Section 12.5...

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Unformatted text preview: (8/5/08) Math 20C. Lecture Examples. Section 12.5. Planes in three space † A vector n = ( a, b, c ) is said to be perpendicular or normal to a plane in xyz-space if it is perpendicular to all lines in the plane (Figure 1). n is perpendicular to all lines in the plane. FIGURE 1 (The point Q = ( x, y, x ) is in the plane if and only if n ·-→ PQ = 0. Theorem 1 The plane through the point P = ( x , y , z ) and perpendicular to the nonzero vector n = ( a, b, c ) has the equation, a ( x- x ) + b ( y- y ) + c ( z- z ) = 0 . (Proof: The point Q = ( x, y, x ) is in the plane if and only if n ·-→ PQ = 0.) Example 1 Give an equation of the plane through the point (2 , 3 , 4) and perpendicular to the vector (- 6 , 5 ,- 4 ) . Answer:- 6( x- 2) + 5( y- 3)- 4( z- 4) = 0 Example 2 Give an equation for the plane through (6 , 10 ,- 3) and perpendicular to the line x =- 3 t, y = 6 + t, z = 4- 7 t ....
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