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If (2a+1)u = (b+1)v + (3c-2)w, what constrains a,b and c if w,u and v are not coplanar?

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islnds | Valedictorian

Posted June 15, 2011 at 10:02 AM via web

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If (2a+1)u = (b+1)v + (3c-2)w, what constrains a,b and c if w,u and v are not coplanar?

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mathsworkmusic | (Level 3) Associate Educator

Posted April 28, 2013 at 7:19 PM (Answer #1)

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Since the planes u, v and w are not coplanar they do not all intersect at any one point (or line, or plane).

This means that when u and v intersect, there is no point on the plane w that lies on this line.

Suppose that u and v intersect on the plane z = r, and values of z on w are denoted z(w) then

(3c - 2)z(w) = r(2a+1-b-1) = r(2a-b)

Since z(w) is never equal to r, this implies that a,b and c are defined by the fact that

`2a-b != 3c-2`   or   `2a-b-3c != -2`

So, if  `b = ma` then `c` cannot equal  `[(2-m)a + 2]/3`  .

Therefore a, b, c can lie anywhere in `RR^3` except on the line

(x,y,z) = (0,0,2/3) + t(1,m,(2-m)/3)

where m = b/a

 

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