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# Insert%20A - SOME PROBLEMS ON GEOMETRY IN SPACE Problem#1...

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SOME PROBLEMS ON GEOMETRY IN SPACE Problem #1 Find the volume of a parallelepiped spanned by vectors u , v and w . Solution : Construct the parallelepiped with adjacent sides u , v and w as shown in the figure below. The area of the parallelogram spanned by v and w is w v × and its height h is h = ϑ cos u (if you consider two cases: 2 0 π < < and < < 2 ). Hence Volume = w v × cos u = ) ( w v u × . Problem #2 Find the equation of the a line passing through the end points of two given vectors a and b . Solution : By “parallelogram” law, the vector a b is parallel to the directed line segment from a to b and we just write the parametric equation of the line passing through a in the direction of . Thus a b ) ( ) ( a b a + = t t l that is b a t t t l + = ) 1 ( ) ( . Note that the last equation for 1 0 t represents the set of all points on the segment from a to b .

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Problem #3 Find the equation of the plane containing the point A( ) and perpendicular to
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Insert%20A - SOME PROBLEMS ON GEOMETRY IN SPACE Problem#1...

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