If a =2 i +3 j +4 k , ~b = i -2 j -2 k , c =- i +4 j +3 k and if d is vector perpendicular to both b and c , a d =18 , then | a d |^2=

Step-by-step solution

Given vectors a = 2 i + 3 j + 4 k , b = i - 2 j - 2 k , and c = - i + 4 j + 3 k . A vector perpendicular to both b and c is given by their cross product: b c = 2 i - j + 2 k . Since d is perpendicular to both b and c , it must be parallel to b c , so d = (2 i - j + 2 k ) for some scalar . Using a d = 18 gives: ((2)(2) + (3)(-1) + (4)(2)) = 18 , (9) = 18 , = 2 . Thus d = 4 i - 2 j + 4 k . The magnitude squared of a d can be found using the identity: | a d |² = | a |²| d |² - ( a d )² . With | a |² = 2² + 3² + 4² = 2