IIT JEE , AIEEE Forum - Mathematics , Vectors

Anupamagrawal13a

ques-ABC is an equilateral triangle of side a.The value of AB.BC+BC.CA+CA+AB is equal to

a)3a*a/2

b)-3a*a/2

Sir the answer is b but sir if put values in the equation we get a as answer.Why.Please give a reason to satisfy me.

iitkgp_bipin

Let position vector of A be 0, that of B be ai and of C be (a/2)i + (a

3/2)j .

AB.BC + BC.CA + CA.AB

= BC.(AB + CA) + CA.AB

= BC.(-BA - AC) + CA.AB

= BC.(-BC) + CA.AB

= -|BC|² + CA.AB

= - | ai - (a/2)i - (a

3/2)j |² + {-(a/2)i - (a

3/2)j}.{ai}

= -a² - a²/2

= -3a²/2

You must have done some minor mistake, use properly the directions of AB,BC,CA.