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Vector Algebra

Question
CBSEENMA12034012
Wired Faculty App

The position vectors of the points A, B and C are 2 space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top comma space space space space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top comma space space space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 3 space straight k with hat on top respectively. Show that A, B and C are collinear.

Solution

Let straight a with rightwards arrow on top space equals space 2 straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top comma space space straight c with rightwards arrow on top space equals space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 3 space straight k with hat on top be position vectors of A, B and C respectively.
     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 equals space open parentheses 3 straight i with hat on top space minus space 2 straight j with hat on top space plus space straight k with hat on top close parentheses space minus space left parenthesis 2 space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis
                                                                                              equals space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top
    AC with rightwards arrow on top space equals space straight P. straight V. space of space straight C space minus space straight P. straight V. space of space straight A space equals space straight c with rightwards arrow on top space minus space straight a with rightwards arrow on top space equals space space open parentheses straight i with hat on top space plus space 4 space straight j with hat on top space minus space 3 space straight k with hat on top close parentheses space minus space left parenthesis 2 straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis 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 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 equals negative space straight i with hat on top space plus space 3 space straight j with hat on top space minus space 2 space straight k with hat on top therefore space space space space AC with rightwards arrow on top space equals space minus AB with rightwards arrow on top therefore space space space space AC with rightwards arrow on top space and space AB with rightwards arrow on top space are space parallel space vectors But space straight A space is space their space common space point. therefore space space space points space straight A comma space straight B comma space straight C space are space collinear.

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