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Vector Algebra

Question
CBSEENMA12034229

Show that the vector area of the triangle ABC whose vertices are 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 is  1 half left parenthesis straight a with rightwards arrow on top space cross times straight b with rightwards arrow on top space plus space straight b with rightwards arrow 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 right parenthesis where 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 vertices A. B and C respectively. Find the condition of collinearity of these points.

Solution
BC 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 B space equals space straight c with rightwards arrow on top minus straight b with rightwards arrow on top
BC 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 B space equals space straight a with rightwards arrow on top minus straight b with rightwards arrow on top
BA with rightwards arrow on top space equals space straight P. straight V. space of space space straight A space minus space straight P. straight V. space of space straight B space equals space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top
BC with rightwards arrow on top space cross times space BA with rightwards arrow on top space equals space left parenthesis straight c with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis
space space space space space space space space space space space space space space space space space space space equals straight c with rightwards arrow on top space cross times space straight a with rightwards arrow on top space minus space straight c with rightwards arrow on top space cross times space straight b with rightwards arrow on top space minus space straight b with rightwards arrow on top space cross times space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top space cross times space straight a with rightwards arrow on top space plus 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 0 with rightwards arrow on top
therefore space space space space BC with rightwards arrow on top space cross times space BA with rightwards arrow on top space equals space straight a with rightwards arrow on top space cross times straight b with rightwards arrow on top space plus space straight b with rightwards arrow 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
therefore space space space space space space vector space area space of space triangle space ABC space equals space 1 half BC with rightwards arrow on top space cross times space BA with rightwards arrow on top
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 space 1 half left parenthesis 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 arrow 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 right parenthesis space space space space space space space space space
If the points A, B and C are collinear, then the vector area of increment ABC = 0 with rightwards arrow on top
 therefore space space space 1 half left parenthesis 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 arrow 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 right parenthesis space equals space 0 with rightwards arrow on top space space space rightwards double arrow space space 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 arrow on top space cross times 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 space equals space 0 with rightwards arrow on top
which is the required condition.