If I haven't already convinced you that your teacher is nothing but a purveyor of falsity, check this out. Let → u be a vector such that | → u | = 1 . Choose a vector → v such that → u ⋅ → v = = 3 and | → v | = √ 5 . Now we have, | → u − → v | = ( → u − → v ) ⋅ ( → u − → v ) = → u ⋅ → u − 2 ( → u ⋅ → v ) + → v ⋅ → v = 0 Hence, → u = → v , since → u − → v = 0 . But → u and → v have different lengths!! Well, gosh darn him anyway! Question #2: How can two things be the same and yet different? Explain.