Chapter 4: Geometry

6. The angle between two planes ¼ Ü · ¼ Ý · ¼ Þ · ¼ ¼and ½ Ü · ½ Ý ·
½ Þ · ½ ¼is
Ó× ½
¼ ½ · ¼ ½ · ¼
Ô Ô
½
(4.12.7)
¾
¼ · ¾ ¼ · ¾ ¼
¾
½ · ¾ ½ · ¾ ½
In particular, the two planes are parallel when ¼ ¼ ¼ ½ ½ ½ , and
perpendicular when ¼ ½ · ¼ ½ · ¼ ½ ¼.
4.12.2 CONCURRENCE AND COPLANARITY
Four planes ¼ Ü· ¼ Ý· ¼ Þ· ¼ ¼, ½ Ü· ½ Ý· ½ Þ· ½ ¼, ¾ Ü· ¾ Ý· ¾ Þ· ¾
¼, and ¿ Ü · ¿ Ý · ¿ Þ · ¿ ¼are concurrent (share a point) if and only if
¼ ¼ ¼ ¼
½ ½ ½ ½
¬ ¾ ¾ ¾ ¾
¬ ¬¬¬¬¬¬¬
¼ (4.12.8)
¬ ¿ ¿ ¿ ¿
Four points ´Ü ¼ Ý ¼ Þ ¼ µ, ´Ü ½ Ý ½ Þ ½ µ, ´Ü ¾ Ý ¾ Þ ¾ µ, and ´Ü ¿ Ý ¿ Þ ¿ µ are coplanar (lie
on the same plane) if and only if
¬
Ü ¼ Ý ¼ Þ ¼ ½
Ü ½ Ý ½ Þ ½ ½
Ü ¾ Ý ¾ Þ ¾ ½
¼ (4.12.9)
Ü ¿ Ý ¿ Þ ¿ ½¬
(Both of these assertions remain true in oblique coordinates.)
¬
4.13 LINES IN SPACE
Two planes that are not parallel or coincident intersect in a straight line, such that
one can express a line by a pair of linear equations
Ü · Ý · Þ · ¼
¼ Ü · ¼ Ý · ¼ Þ · ¼ (4.13.1)
¼
such that ¼ ¼ , ¼ ¼ , and ¼ ¼ are not all zero. The line thus defined is
parallel to the vector ´ ¼ ¼ , ¼ ¼ , ¼ ¼ µ. The direction cosines of the line
are those of this vector. See Equation (4.11.1). (The direction cosines of a line are
only defined up to a simultaneous change in sign, because the opposite vector still
gives the same line.)
The following particular cases are important:
1. Line through ´Ü ¼ Ý ¼ Þ ¼ µ parallel to the vector ´ µ:
Ü Ü ¼
Ý Ý ¼
Þ Þ ¼
(4.13.2)
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