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Vector Algebra

Question
CBSEENMA12034055
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Given two points A and B, identify the set of points P in space such that PA with rightwards arrow on top. space PB with rightwards arrow on top space less than space 0.

Solution

We have
                  PA with rightwards arrow on top. space PB with rightwards arrow on top space less than space 0
 therefore space space left parenthesis PA right parenthesis space left parenthesis PB right parenthesis space space cos space angle APB space equals space 0 rightwards double arrow space space space cos space angle APB thin space less than thin space 0 space space space space space space space space space space space space space space open square brackets because PA greater than 0 comma space PB greater than 0 close square brackets rightwards double arrow space space space angle APB space is space obtuse space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis

Draw a sphere with O as centre and AB as diameter. Take any point P in the interior of this sphere and join PA and PB. It is clear that P is an interior point of the sphere if and only if ∠APB is obtuse.
∴  from (1), it follows that the required set of points is the interior of the sphere with AB as a diameter.

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