aligned & If a =4 i +3 j + k , b = i -2 j +2 k & then a ( a ( a ( a b )))= aligned

Step-by-step solution

Given vectors a = 4 i + 3 j + k and b = i - 2 j + 2 k , the dot product is a b = (4)(1) + (3)(-2) + (1)(2) = 0 , confirming orthogonality. The magnitude squared is | a |^2 = 4^2 + 3^2 + 1^2 = 26 . Evaluating the nested cross product a ( a ( a ( a b ))) Using the vector triple product identity x ( y z ) = ( x z ) y - ( x y ) z : Let v ₂ = a ( a b ) = ( a b ) a - ( a a ) b = -26 b . Then v ₃ = a v ₂ = a (-26 b ) = -26( a b ) . Finally, v ₄ = a v ₃ = a (-26 v ₁) = -26( a v ₁) = -26(-26 b ) = 676 b . The result simplif