The line AB is a common perpendicular to to two skew lines AP and BQ, and C and R are the midpoints of AB and PQ respectively. Prove by vector methods, that CR and AB are perpendicular.
I tried the vectors, AP=s, AB=b, BQ=c, and PQ=d
a.b=0, and b.c=0
CR=
then AB.CR=
this would equal 0 if b.d is equal to one but I don't see how that relationship comes about.
Thanks!
1. LetOriginally Posted by arze
The line AB is a common perpendicular to to two skew lines AP and BQ, and C and R are the midpoints of AB and PQ respectively. Prove by vector methods, that CR and AB are perpendicular.
I tried the vectors, AP=s, AB=b, BQ=c, and PQ=d
a.b=0, and b.c=0
CR=
then AB.CR=
this would equal 0 if b.d is equal to one but I don't see how that relationship comes about.
Thanks!denote the stationary vector of point A,
the stationary vector of point B, etc..
2. According to the text you know:
3. The stationary vector of point C is calculated by:
The stationary vector of point R is calculated by:
Consequently the direction vector
4. Now you have to calculate:
 = \frac12(\overrightarrow{AB} \cdot \overrightarrow{AP} + \overrightarrow{AB} \cdot \overrightarrow{BQ}) = \frac12(0+0)=0)
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