3. Let a = (1,0, 1, 2) and b 3D (-,,,-). 1 1 1 2' 2'...

    

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3. Let a = (1,0, 1, 2) and b 3D (-,글,글,-). 1 1 1 2' 2' 2? 3 1. Use Theorem 1.3.2 (i) to find Proj→ 2. Use Theorem 1.3.2 O (ii) to find 0, where e is the angle between a and b. Theorem 1.3.2. Let a, b ER" be nonzero vectors. Then the following assertions hold. a i. ||Proj 6 Proj- b where 0 is the angle between a and b ii. cos 0| %3D Proof i. Because Proj, b = ( a· b a, we have a b a a ||Proj- b || =|| || á || 12 ellp|| ii. By (1.2.15) O and (i), we have ||Proj: |cos 0| = and the result (ii)holds. By Theorem 1.3.2 D, we see that the following assertions hold. Proj; 7| 1. If 0 E (0, 5) , then cos 0 |Proj- b 2. If 0 E (5, 7), then cos 0 a 3. Let a = (1,0, 1, 2) and b 3D (-,글,글,-). 1 1 1 2' 2' 2? 3 1. Use Theorem 1.3.2 (i) to find Proj→ 2. Use Theorem 1.3.2 O (ii) to find 0, where e is the angle between a and b. Theorem 1.3.2. Let a, b ER" be nonzero vectors. Then the following assertions hold. a i. ||Proj 6 Proj- b where 0 is the angle between a and b ii. cos 0| %3D Proof i. Because Proj, b = ( a· b a, we have a b a a ||Proj- b || =|| || á || 12 ellp|| ii. By (1.2.15) O and (i), we have ||Proj: |cos 0| = and the result (ii)holds. By Theorem 1.3.2 D, we see that the following assertions hold. Proj; 7| 1. If 0 E (0, 5) , then cos 0 |Proj- b 2. If 0 E (5, 7), then cos 0 a