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Vector Algebra

Question
CBSEENMA12034013
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The position vectors of the points A, B, C and D are 4 straight i with hat on top plus 3 space straight j with hat on top space minus space straight k with hat on top comma space space 5 space straight i with hat on top plus space 2 space straight j with hat on top space plus space 2 space straight k with hat on top comma space space 2 space straight i with hat on top space minus space space 2 space straight j with hat on top space minus space 3 space straight k with hat on top space and space 4 space straight i with hat on top space minus space space 4 space straight j with hat on top space plus space 3 space straight k with hat on top respectively. Show that AB and CD are parallel.

Solution

Let 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 be P.V s. of A, B, C and D respectively.
therefore space space space space straight a with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space space 3 space straight j with hat on top space minus space straight k with hat on top comma space space space space space space space space space space straight b with rightwards arrow on top space equals space 5 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 2 space straight k with hat on top space space space space space space space space space straight c with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 2 space straight j with hat on top space minus space space 3 space straight k with hat on top comma space space space space space space straight d with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space 4 space straight j with hat on top space plus 3 space straight k with hat on top
AB with rightwards arrow on top space equals space straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight A space equals space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top space space space space space space space equals space left parenthesis 5 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis space minus space left parenthesis 4 space straight i with hat on top space plus space 3 space straight j with hat on top space minus space straight k with hat on top right parenthesis space equals space straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top
CD with rightwards arrow on top space equals space straight P. straight V. space of space straight D space minus space straight P. straight V. space of space straight C space equals space straight d with rightwards arrow on top space minus straight c with rightwards arrow on top space space space space space space space equals space left parenthesis 4 space straight i with hat on top space minus space 4 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space minus space left parenthesis 2 space straight i with hat on top space minus 2 space straight j with hat on top space minus space 3 space straight k with hat on top right parenthesis space equals space 2 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 6 space straight k with hat on top space equals space 2 space left parenthesis straight i with hat on top space minus straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis therefore space space space CD with rightwards arrow on top space equals space 2 space AB with rightwards arrow on top space space space space rightwards double arrow space space space space AB with rightwards arrow on top space space and space CD with rightwards arrow on top space are space parallel.

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