THE SCALAR EQUATION OF A PLANE In general can the equation of a plane be given

# The scalar equation of a plane in general can the

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THE SCALAR EQUATION OF A PLANE In general can the equation of a plane be given as: ³ 3 + ´ 4 + ² 5 = 8 And the vector ( a , b , c ) is perpendicular to the plane
9/2/2015 4 THE SCALAR EQUATION OF A PLANE We can then write the scalar equation of the plane as: µ − µ : . 0 = 0 With: µ = +,-./.,0 12²/,µ = 3, 4, 5 µ : = ³ +,.0/ /6³/ 7.2- ,0 /62 +7³02 e. g. = x : , y : , z : 0 = /62 12²/,µ +2µ+208.²97³µ /, /62 +7³02 = (0 @ , 0 A , 0 B ) Thus: µ − µ : . 0 = 0 3, 4, 5 3 : , 4 : , 5 : . 0 @ , 0 A , 0 B = 0 0 @ 3 − 3 : + 0 A 4 − 4 : + 0 B 5 − 5 : = 0 THE SCALAR EQUATION OF A PLANE µ − µ : . 0 = 0 3, 4, 5 3 : , 4 : , 5 : . 0 @ , 0 A , 0 B = 0 0 @ 3 − 3 : + 0 A 4 − 4 : + 0 B 5 − 5 : = 0 0 @ 3 − 0 @ 3 : + 0 A 4 − 0 A 4 : + 0 B 5 − 0 B 5 : = 0 0 @ 3 + 0 A 4 + 0 B 5 = 0 @ 3 : + 0 A 4 : + 0 B 5 : In the same format as: ³3 + ´3 + ²4 = 8
9/2/2015 5 EXAMPLES Example 2. The two vectors ³ = 1 2 2 and b = 1 −2 1 are both in a certain plane. The point (1,1,0) is also in the plane. (a) Find the vector equation of the plane . (b) Find the parametric equations of the plane. (c) Find the equation of the plane in terms of C, D and E only. Solution: (a): µ = 3 4 5 = 1 1 0 + ¸ 1 2 2 + ¹ 1 −2 1 . (b): 3 = 1 + ¸ + ¹ 4 = 1 + 2¸ − 2¹ 5 = 2¸ + ¹ (c) Find the equation of the plane in terms of C, D and E only. Solution: Parametric equations are 3 = 1 + ¸ + ¹ (1) 4 = 1 + 2¸ − 2¹ (2) 5 = 2¸ + ¹ (3) (1): ¸ + ¹ = 3 − 1 (3): 2¸ + ¹ = 5 Subtract: ¸ = 5 − 3 + 1.

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