Unformatted text preview: or x−2 y−1 z−4
−1
−2
−2
2
−2
−3 = 2x − 7y + 6z − 21 = 0. 16. Non–parallel lines L and M in three dimensional space are given by
equations
P = A + sX, Q = B + tY.
E (i) Suppose P Q is orthogonal to both X and Y . Now
E E P Q= Q − P = (B + tY ) − (A + sX ) =AB +tY − sX.
Hence
E (AB +tY + sX ) · X = 0
E (AB +tY + sX ) · Y = 0. More explicitly
E t(Y · X ) − s(X · X ) = − AB ·X
E t(Y · Y ) − s(X · Y ) = − AB ·Y.
However the coeﬃcient determinant of this system of linear equations
in t and s is equal to
Y · X −X · X
Y · Y −X · Y = −(X · Y )2 + (X · X )(Y · Y )
= X × Y 2 = 0, as X = 0, Y = 0 and X and Y are not proportional (L and M are
not parallel).
(ii) P and Q can be viewed as the projections of C and D onto the line P Q,
where C and D are arbitrary points on the lines L and M, respectively.
Hence by equation (8.14) of Theorem 8.5.3, we have
P Q ≤ CD.
Finally we derive a useful formula for P Q. Again by Theorem 8.5.3
E E E
 AB · P Q 
ˆ
=  AB ·n,
PQ =
PQ 94 ...
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 Fall '10
 Dreibelbis
 Elementary algebra

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