MATH 210 Self-quiz 8: Determining Angles and Orthogonal Vectors, Quizzes of Advanced Calculus

Download MATH 210 Self-quiz 8: Determining Angles and Orthogonal Vectors and more Quizzes Advanced Calculus in PDF only on Docsity! MATH 210 Self-quiz 8 1. Determine whether the angle between the vectors 〈3, 2,−7〉 and 〈1, 1, 1〉 is acute, right or obtuse. 2. Show that if u, v are two nonzero vectors with: |u|2 + |v|2 = |u+ v|2 then the two vectors are orthogonal. MATH 210 Self-quiz 8 1. Determine whether the angle between the vectors 〈3, 2,−7〉 and 〈1, 1, 1〉 is acute, right or obtuse. Solution: The dot product of these two vectors is: 3 · 1 + 2 · 1 + (−7) · 1 = −2 Since the sign of the dot product is the same as the sign of the cosine of the angle between them, we must have: cos θ < 0 where θ is the angle between the two vectors. This means that this angle is obtuse. 2. Show that if u, v are two nonzero vectors with: |u|2 + |v|2 = |u+ v|2 then the two vectors are orthogonal. Solution: We recall that: • |u|2 = u · u • |v|2 = v · v • |u+ v|2 = (u+ v) · (u+ v) This means that the given relation translates to: