MATH 32A BRIDGE 2013 M Wang Homework 3 Solution 1 Compute the dot product 13 3 Exercises 1 10 a cid 104 1 2 1 cid 105 cid 104 4 3 5 cid 105 Ans cid 104 1 2 1 cid 105 cid 104 4 3 5 cid 105 4 6 5 15 b 3j 2k i 4k Ans 3j 2k i 4k 3i j 12j k 2i k 8k k 0 0 0 8 8 2 Find the cosine of the angle between the vectors 13 3 Exercises 19 22 a cid 104 0 3 1 cid 105 cid 104 4 0 0 cid 105 Ans cos cid 104 0 3 1 cid 105 cid 104 4 0 0 cid 105 cid 107 cid 104 0 3 1 cid 105 cid 107 cid 107 cid 104 4 0 0 cid 105 cid 107 0 b 3i k i j k Ans cos 3i k i j k cid 107 3i k cid 107 cid 107 i j k cid 107 4 10 3 2 15 30 3 Simplify the expression 13 3 Exercises 35 37 a v w v v w Ans v w v v w v2 v w v w v2 b v w v v w w Ans v w v v w w v2 w v v w w2 v2 w2 4 Show that if e and f are unit vectors such that cid 107 e f cid 107 3 e f 1 8 13 3 Exercises 47 Proof Since e and f are unit vectors we know that e e cid 107 e cid 107 2 f f cid 107 e cid 107 2 1 cid 107 e f cid 107 3 2 e f 2 2e f 1 4 cid 107 e f cid 107 7 2 e2 f 2 2e f 7 4 9 4 e2 f 2 2e f 9 4 5 Find the projection of u along v 13 3 Exercises 51 55 1 2 then cid 107 e f cid 107 2 Hint Show that 7 a u cid 104 2 5 cid 105 v cid 104 1 1 cid 105 v Ans u cid 107 b u 5i 7j 4k v k Ans u cid 107 v cid 17 cid 17 cid 16 u v cid 16 u v v v v v cid 16 7 cid 17 cid 104 1 1 cid 105 cid 104 7 cid 17 cid 16 4 k 4k 2 1 2 cid 105 2 7 6 Find the decomposition a a cid 107 a with respect to b 13 3 Exercises 64 65 a a cid 104 2 3 cid 105 b cid 104 5 0 cid 105 Ans a cid 107 10 b a cid 104 4 1 0 cid 105 b cid 104 0 1 1 cid 105 25 cid 104 5 0 cid 105 cid 104 2 0 cid 105 a a a cid 107 cid 104 0 3 cid 105 2 cid 104 0 1 1 cid 105 cid 104 0 1 2 1 2 cid 105 a a a cid 107 cid 104 4 1 2 cid 105 2 1 Ans a cid 107 1 2