Homework SourseIf A a11 a12 a21 a22 and B b11 b12 b21 b22 are arbitrary v
If A = [a_11 a_12 a_21 a_22] and B = [b_11 b_12 b_21 b_22] are arbitrary vectors in R^2 times 2, then the mapping (A, B) = a_11 b_11 + a_12 b_12 + a_21 b_21 + a_22b_22 defines an inner product in R^2 times 2. Use this inner product to determine (A, B), ||A||, ||B||, and the angle a_A, B between A and B for (A, B) = ||A|| = ||B|| = a_A, B = ![If A = [a_11 a_12 a_21 a_22] and B = [b_11 b_12 b_21 b_22] are arbitrary vectors in R^2 times 2, then the mapping (A, B) = a_11 b_11 + a_12 b_12 + a_21 b_21 +](/WebImages/35/if-a-a11-a12-a21-a22-and-b-b11-b12-b21-b22-are-arbitrary-v-1105532-1761585026-0.webp "If A = [a_11 a_12 a_21 a_22] and B = [b_11 b_12 b_21 b_22] are arbitrary vectors in R^2 times 2, then the mapping (A, B) = a_11 b_11 + a_12 b_12 + a_21 b_21 +")
Solution
<A, B> = a11b11 + a12b12+ a21b21 = -16 -5 -10 -5 = -46
||A||^2 = <A, A> = 16 + 1+ 25 + 25 = 67
||A|| = sqrt67
||B||^2 = <B, B> = 16 + 25 + 4 + 1 = 46
||B|| = sqrt46
cosalpha = <A B >/||A||||B|| = -46/sqrt(67*46) = -46/55.15 = -0.834
alpha = 146.51 deg
