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Unformatted text preview: Hence the shortest distance between lines AC and BD is equal to 0
−6 −2 · 1 E
1
5
3
 AB ·(X × Y )
√
=
=√ .
X × Y 
62
62
18. Let E be the foot of the perpendicular from A4 to the plane A1 A2 A3 .
Then
1
vol A1 A2 A3 A4 = ( area ∆A1 A2 A3 ) · A4 E.
3
Now
E
E
1
area ∆A1 A2 A3 =  A1 A2 × A1 A3 .
2
Also A4 E is the length of the projection of A1 A4 onto the line A4 E . (See
ﬁgure above.)
E Hence A4 E =  A1 A4 ·X , where X is a unit direction vector for the line
A4 E . We can take
E
E
A1 A2 × A1 A3
X=
.
E
E
 A1 A2 × A1 A3 
Hence E vol A1 A2 A3 A4 =
= E E E
E
1
 A1 A4 ·(A1 A2 × A1 A3 )
 A1 A2 × A1 A3 
E
E
6
 A A × A A 
E
E
E
1
 A1 A4 ·(A1 A2 × A1 A3 )
6 96 1 2 1 3 ...
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This note was uploaded on 12/19/2011 for the course MAS 3105 taught by Professor Dreibelbis during the Fall '10 term at UNF.
 Fall '10
 Dreibelbis

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