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Vector Algebra

Question
CBSEENMA12034232
Wired Faculty App

 Let A, B and C be any three non-collinear points with position vectors 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 respectively. Show that the perpendicular distance from C to the straight line through A and B is fraction numerator open vertical bar 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 space plus space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar over denominator open vertical bar straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top close vertical bar end fraction.

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 be position vectors of A, B, C respectively.
From C, draw CL space perpendicular space AB.
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 straight c with rightwards arrow on top space minus space straight a with rightwards arrow on top
In rt. angle straight d   increment ALC comma
                CL over AC space equals sin space straight A
therefore space space space space CL space equals space AC space sin space straight A

                  equals space fraction numerator left parenthesis AC right parenthesis thin space left parenthesis AB right parenthesis space sin space straight A over denominator AB end fraction space equals space fraction numerator open vertical bar AC with rightwards arrow on top cross times space AB with rightwards arrow on top close vertical bar over denominator open vertical bar AB with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator open vertical bar left parenthesis straight c with rightwards arrow on top space minus straight a with rightwards arrow on top right parenthesis space cross times left parenthesis straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top right parenthesis close vertical bar over denominator open vertical bar straight b with rightwards arrow on top minus straight a with rightwards arrow on top close vertical bar end fraction equals space fraction numerator open vertical bar straight c with rightwards arrow on top space cross times space straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top cross times space stack straight a space with rightwards arrow on top space minus space straight a with rightwards arrow on top space cross times space straight b 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 close vertical bar over denominator open vertical bar straight b with rightwards arrow on top minus space straight a with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator open vertical bar negative straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top space minus space straight c with rightwards arrow on top space cross times straight a with rightwards arrow on top space minus space straight a with rightwards arrow on top cross times space straight b with rightwards arrow on top space plus space 0 with rightwards arrow on top close vertical bar over denominator open vertical bar straight b with rightwards arrow on top space minus straight a with rightwards arrow on top close vertical bar end fraction therefore space space space required space distance space equals space fraction numerator open vertical bar 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 close vertical bar over denominator open vertical bar straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top close vertical bar end fraction.

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