(6) show that u = (13)(2, 2,1) and v = (13)(1,2,2) are orthogo

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(6) show that u = (1/3)(2,-2,1) and v = (1/3)(1,2,2) are

orthogonal unit vectors parallel to the plane 2x + y - 2z = -

2, and that P(1,2,3) and Q(-2,4,1) are points in the plane.

hence find parameters s and t such that:

q = (-2,4,1) = (1,2,3) + s(2/3, -2/3, 1/3) + t(1/3,

2/3, 2/3).

Solution

orthogonal => a.b = 0

u.(2x+y-2z) =

(2/3)(2)+(-2/3)(1)+(1/3)(-2)

= 0

v.(2x+y-2z)= (1/3)(2) + (2/3)(1) + (2/3)(-2) = 0

substitute P in plane eq

2(1)+2-2(3)+2

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(6) show that u = (1/3)(2,-2,1) and v = (1/3)(1,2,2) are orthogonal unit vectors parallel to the plane 2x + y - 2z = 2, and that P(1,2,3) and Q(-2,4,1) are points in the plane. hence find parameters s and t such that: q = (-2,4,1) = (1,2,3) + s(2/3, -2/3, 1/3) + t(1/3, 2/3, 2/3). Solution orthogonal => a.b = 0 u.(2x+y-2z) = (2/3)(2)+(-2/3)(1)+(1/3)(-2) = 0 v.(2x+y-2z)= (1/3)(2) + (2/3)(1) + (2/3)( -2) = 0 substitute P in plane eq 2(1)+2-2(3)+2 = 0 hence in plane substitute Q 2(-2)+4-2(1)+2 = 0 Hence in plane -2 = 1+ 2s/3 + t/3 2s+t = -9 ------ (1) 4 = 2 - 2s/3 +2t/3 -2s+2t = 6 ---- (2) ...
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