(A B C ) is a right-angled triangle in which ( A B, B C, A C =B C ). If the position vectors of (B ) and (C ) are respectively (3 i -2 j + k ) and (5 i + j -3 k ), then [ A B A C + B A B C + C A C B = ]

Step-by-step solution

Given ( aligned B & =3 i -2 j + k C & =5 i + j -3 k B C & =2 i +3 j -4 k aligned ) ( A B, B C, A C =B C ) ( B C ) is hypotenuse of ( A B C ) ( aligned & A =90^ & A B A C =0 & B A B C =| B A | B C | B & C A C B =| C A | C B | C & A B - A C + B A - B C + C A - C B &=0+| B C |(| B A | B+| C B | C) &=0+| B C | B C | [ By projection formula ] &=| B C |^2- ( (2)^2+3^2+4^2 )^2=4+9+16=29 aligned )