Let a = 2 i + k , b = i + j + k , and c = 4 i - 3 j + 7 k . If r is a vector such that r b = c b and r a = 0 , Then r c =

Step-by-step solution

Given r b = c b ( r - c ) b = 0 r - c = b r = c + b Taking the dot product with a : r a = c a + ( b a ) = 0 Calculating the dot products: c a = (4)(2) + (-3)(0) + (7)(1) = 15 b a = (1)(2) + (1)(0) + (1)(1) = 3 Substituting these values: 15 + 3 = 0 = -5 Thus, r = c - 5 b r = (4 i - 3 j + 7 k ) - 5( i + j + k ) = - i - 8 j + 2 k Now, calculating r c : r c = (- i - 8 j + 2 k ) (4 i - 3 j + 7 k ) r c = -4 + 24 + 14 = 34 Answer: 34