Lecture 33 Line in R 2x y z 6 and x y 2z 3 Normals 2 1 1 and 1 1 2 To parameterize we need two points z 0 2x y 6 x y 3 3 0 0 z 0 y z 6 y 2z 3 0 3 3 Direction 3 0 0 0 3 3 3 3 3 r t 3 t t t Now introduce another line r s 4 5s 1 s 2 s What is the shortest distance between the two lines De ne f t s distance r t r s F 3 t 4 5s t 1 s t 2 s Critical points f 2 8 3t 5s and f 2 32 5t 27s We want to solve 3t 5s 8 and 5t 27s 32 r 2 1 1 and r 1 2 3 Distance 14 Find the tangent plane to 2x 4y z 1 at 1 0 1 V 2x 4y z 4x 8y 2z is normal at 1 0 1 that is 4 0 2 4x 2z d d 2 multiplier Find the closest point to 1 2 3 on the unit sphere centered at zero a 1 2 3 a 4a 9a 1 14a 1 We get that the best point is We want to minimize f x y z x 1 y 2 z 3 such that g x y z x y z 1 We want to minimize Vf Vg and g 1 x 1 x y 2 y z 3 z 1 x y z 1 4 9 14 1 14 so x y z
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