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Vector Algebra

Question
CBSEENMA12033962
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Show that the points negative 2 space straight a with rightwards arrow on top space plus space 3 space straight b with rightwards arrow on top space plus space 5 space straight c with rightwards arrow on top comma space space straight a with rightwards arrow on top space plus space 2 space straight b with rightwards arrow on top space plus space 3 space straight c with rightwards arrow on top comma space space 7 space straight a with rightwards arrow on top space minus space straight c with rightwards arrow on top are collinear, where straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space and space straight c with rightwards arrow on top are three non-coplanar vectors.

Solution

Let negative 2 space straight a with rightwards arrow on top space plus space 3 space straight b with rightwards arrow on top space plus space 5 space straight c with rightwards arrow on top comma space space straight a with rightwards arrow on top space plus space 2 space straight b with rightwards arrow on top space plus space 3 space straight c with rightwards arrow on top comma space space 7 space straight a with rightwards arrow on top space minus space straight c with rightwards arrow on top be the 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 open parentheses straight a with rightwards arrow on top space plus space 2 straight b with rightwards arrow on top space plus space 3 straight c with rightwards arrow on top close parentheses space minus space open parentheses negative 2 straight a with rightwards arrow on top space plus space 3 straight b with rightwards arrow on top space plus space 5 straight c with rightwards arrow on top close parentheses space equals space 3 straight a with rightwards arrow on top minus straight b with rightwards arrow on top minus 2 straight c with rightwards arrow on top
Also, 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 open parentheses 7 straight a with rightwards arrow on top space minus space straight c with rightwards arrow on top close parentheses space minus space open parentheses negative 2 straight a with rightwards arrow on top space plus space 3 straight b with rightwards arrow on top space plus space 5 straight c with rightwards arrow on top close parentheses space equals space 9 straight a with rightwards arrow on top minus straight b with rightwards arrow on top minus 2 straight c with rightwards arrow on top
                                           equals space 3 space open parentheses 3 straight a with rightwards arrow on top space minus straight b with rightwards arrow on top space minus space 2 space straight c with rightwards arrow on top close parentheses
therefore space space space space AC with rightwards arrow on top space equals 3 space AB with rightwards arrow on top rightwards double arrow space space space space AB with rightwards arrow on top space and space AC with rightwards arrow on top space are space parallel space vectors
But the vectors have the same initial point point A, B and C are collinear.

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