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Vector Algebra

Question
CBSEENMA12034039
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Four points A, B, C, D with position vectors straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top comma space straight d with rightwards arrow on top respectively are such that 3 straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space plus space space 2 straight c with rightwards arrow on top space minus space 4 space straight d with rightwards arrow on top space equals space 0 with rightwards arrow on top. Show that the four points are coplanar. Also, find the position vector of the point of intersection of lines AC and BD.

Solution

Since,       3 space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space plus space 2 space straight c with rightwards arrow on top space minus space 4 space straight d with rightwards arrow on top space equals space 0 with rightwards arrow on top
therefore space space 3 space straight a with rightwards arrow on top space plus space 2 space straight c with rightwards arrow on top space equals space straight b with rightwards arrow on top space plus space 4 space straight d with rightwards arrow on top space space space rightwards double arrow space space fraction numerator 3 space straight a with rightwards arrow on top space plus space 2 space straight c with rightwards arrow on top over denominator 5 end fraction space equals fraction numerator straight b with rightwards arrow on top space plus space 4 space straight d with rightwards arrow on top over denominator 5 end fraction space rightwards double arrow space space fraction numerator 3 space straight a with rightwards arrow on top space plus space 2 space straight c with rightwards arrow on top over denominator 3 plus 2 end fraction space equals space fraction numerator straight b with rightwards arrow on top space plus 4 space straight d with rightwards arrow on top over denominator 1 plus 4 end fraction

⇒ P.V. of point dividing AC in the ratio 2 : 3 is the same as the P.V. of point dividing BD in the ratio 4 : 1
∴  AC and BD intersect at P i.e., A, B, C, D are coplanar and P has P.V. as
fraction numerator 3 straight a with rightwards arrow on top space plus space 2 straight c with rightwards arrow on top over denominator 5 end fraction space space or space space fraction numerator straight b with rightwards arrow on top space plus space 4 space straight d with rightwards arrow on top over denominator 5 end fraction

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