Let a = (a₁ i + a₂ j + a₃ k ), b = (b₁ i + b₂ j + b₃ k ), c = (c₁ i + c₂ j + c₃ k ) be three non-zero vectors such that a is a unit vector perpendicular to both b and c . If the angle between b and c is 3 then vmatrix a₁ & a₂ & a₃ b₁ & b₂ & b₃ c₁ & c₂ & c₃ vmatrix ^2 =

Step-by-step solution

The given determinant represents the scalar triple product of the vectors a , b , and c . vmatrix a₁ & a₂ & a₃ b₁ & b₂ & b₃ c₁ & c₂ & c₃ vmatrix = [ a b c ] = a ( b c ) Since a is perpendicular to both b and c , the vector a is collinear with b c . [ a b c ] = | a | | b c | Given that a is a unit vector, | a | = 1 . [ a b c ]^2 = | b c |^2 The angle between b and c is 3 . | b c |^2 = (| b | | c | ( 3 ) )^2 | b c |^2 = | b |^2 | c |^2 ( 3 2 )^2 = 3 4 | b |^2| c |^2 Answer: 3 4 | b |^2| c |^2