Let a = 7 i + j - k and b = j + 2 k . If r is a vector such that r a + a b = 0 and r a = 0 , then |3 r |^2 is equal to:

Step-by-step solution

Given r a + a b = 0 r a - b a = 0 ( r - b ) a = 0 r - b = a r = b + a Taking dot product with a on both sides: r a = b a + | a |^2 Since r a = 0 , we get: = - b a | a |^2 We have a = 7 i + j - k and b = j + 2 k . b a = (0)( 7 ) + (1)(1) + (2)(-1) = -1 | a |^2 = ( 7 )^2 + (1)^2 + (-1)^2 = 9 | b |^2 = (1)^2 + (2)^2 = 5 Substituting these values: = - -1 9 = 1 9 Therefore, r = b + 1 9 a Squaring both sides: | r |^2 = | b |^2 + 1 81 | a |^2 + 2 9 ( b a ) | r |^2 = 5 + 1 81 (9) + 2 9 (-1) | r |^2 = 5 + 1 9 - 2 9 = 44 9 F