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Question

If the vertices of a triangle are A(2,1,1), B(1,3,5) and C(3,4,4), then prove by vector method that it is a right-angled triangle.

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Solution

A=2^I^J+^kB=^I3^J5^kC=3^I+4^J4^kAB=BA=(^I3^J5^k)(2^I^J+^k)=^I2^J6^kBC=CB=(3^I+4^J4^k)(^I3^J5^k)=2^I+7^J+^kAC=CA=(3^I+4^J4^k)(2^I^J+^k)=^I+5^J5^ktwovectorsareprpendiculartoeachother,iftheirscalarproductiszeroAB.BC=(^I2^J6^k).(2^I+7^J+^k)=2146=22BC.AC0AB.AC0no2vectorsareprependiculartoeachothercoordinatesofCmustbe(3,4,4)toprovethetriangleisrightangled.

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