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Vector Algebra

Question
CBSEENMA12034231
Wired Faculty App

If 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 are the position vectors of the non-collinear points A, B, C respectively in space, show that straight b with rightwards arrow on top cross times straight c with rightwards arrow on top space plus space straight c with rightwards arrow on top space cross times straight a with rightwards arrow on top space plus space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top is perpendicular to plane ABC.

Solution

We have,
              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 minus straight a with rightwards arrow 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
Vector perpendicular to plane ABC
                          = Vector perpendicular to AB with rightwards arrow on top space and space AC with rightwards arrow on top
                          equals space AB with rightwards arrow on top cross times AC with rightwards arrow on top space equals space left parenthesis straight b with rightwards arrow on top minus straight a with rightwards arrow on top right parenthesis space cross times space left parenthesis straight c with rightwards arrow on top minus straight a with rightwards arrow on top right parenthesis equals space straight b with rightwards arrow on top cross times straight c with rightwards arrow on top space minus straight b with rightwards arrow on top space cross times space straight a with rightwards arrow on top space minus space straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top space plus space straight a with rightwards arrow on top cross times space straight a with rightwards arrow on top space equals space straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top space plus space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top space cross times space straight a with rightwards arrow on top space plus space 0 with rightwards arrow on top equals space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space plus space straight b with rightwards harpoon with barb upwards on top space cross times space straight c with rightwards arrow on top space plus space straight c with rightwards arrow on top space cross times space straight a with rightwards arrow on top

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