Question

Find a unit vector which should lie on the plane determined by the vectors 2 i + j + k & i + 2 j + k and perpendicular to i + j + 2k.

Answer 1

Let vector a = 2i + j + k = ( 2 , 1 , 1 )

vector b = i + 2j + k = ( 1 , 2 , 1)

vector c = i + j + 2k = (1 , 1 , 2)

consider,

vector (a x b ) x c) = vector (c ● a) b –vector(c ● b ) a ---------- ( 1) 

vector (c ● a) = 2 + 1 + 2 = 5 

vector (a x b) x c) = 5 vector b – 5 vector a

= 5 vector (b – a) 

= 5 vector (-1 , 1 , 0)

|vector (a x b ) x c| = 5 √ ( ! + 1 + 0 ) = 5√2 units 

η = unit vector coplanar with a & b and perpendicular to c