Let a and b be linearly independent vectors such that | a | = 3 , | b | = 3 and | a - b | = 4 . If a (2 i + 2 j - k ) = (2 i + 2 j - k ) b and |( a + b ) (3 i + 4 j + 2 k )| = , then =

Step-by-step solution

Given | a - b | = 4 , squaring both sides gives: | a |^2 + | b |^2 - 2 a b = 16 Substituting | a | = 3 and | b | = 3 : 3 + 9 - 2 a b = 16 2 a b = -4 Now, calculating | a + b |^2 : | a + b |^2 = | a |^2 + | b |^2 + 2 a b = 3 + 9 - 4 = 8 Let c = 2 i + 2 j - k . The given cross product relation is: a c = c b a c = - b c ( a + b ) c = 0 This implies that a + b is parallel to c . Thus, a + b = t c for some scalar t . Taking the magnitude on both sides: | a + b | = |t| | c | Since | c | = 2^2 + 2^2 + (-1)^2 = 3 and | a +