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Explain geometrically why (x,y,z)=(1,2,3)+s(1,1,1)+t(-2,-2,-2) is not the equation of a plane but rather the equation of a line. Show algebraically that this is the equation of a line (in the form (x,y,z)=(a1,a2,a3)+n(m1,m2,m3)

 Jun 16, 2020
 #1
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(1, 1, 1) and (-2, -2, -2) are linearly dependent vectors, which means all points on (x, y, z) = (1, 2, 3) + s(1, 1, 1) + t(-2, -2, -2) all satisfies (x - 1, y - 2, z - 3) = (s - 2t) (1, 1, 1), which means it is a straight line passing through the point (1, 2, 3).

 

(x, y, z) = (1, 2, 3) + s(1, 1, 1) + t(-2, -2, -2)

(x, y, z) = (1, 2, 3) + s(1, 1, 1) - 2t(1, 1, 1)

(x, y, z) = (1, 2, 3) + (s - 2t) (1, 1, 1)

It is an equation of a line.

 Jun 17, 2020

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