Comment. Inner products quiz Comment. This quiz is designed to test your knowledge of inner product spaces and related concepts Comment. such as inner products, length, orthogonality, and orthonormal bases. Shuffle Questions. Shuffle Answers. Question. Let V be an inner product space, and let x and y be two vectors in V such that || x || = 3 and || y || = 4. What exactly can we say about || x + y ||? Answer. || x + y || = 5. Comment. This is true if x and y are orthogonal (by Pythagoras's theorem), but is not true otherwise. Answer. || x + y || is less than or equal to 5. Partially Correct Answer. || x + y || is less than or equal to 7. Correctness. 3 Comment. It is true that || x + y || is less than or equal to 7 (by the triangle inequality), but this is not the only thing one can say about || x + y ||. Partially Correct Answer. || x + y || is between 0 and 7 inclusive. Comment. It is true that || x + y || is less than or equal to 7 (by the triangle inequality), and must be at least 0 (by positivity), but this is not the only thing one can say about || x + y ||. Answer. || x + y || is equal to 1 or 7. Comment. It is also possible for || x + y || to take values between 1 and 7. Remember that x and y are _vectors_, not _numbers_; saying that || x || = 3 does not mean that x is equal to +3 or -3, and similarly for y. Correct Answer. || x + y || is between 1 and 7 inclusive. Answer. || x + y || is equal to 7. Question. Let V be a complex inner product space, and let x and y be two vectors in V such that || x || = 3 and || y || = 4. What exactly can we say about < x, y >? Answer. < x, y > = 0. Comment. This is true if x and y are orthogonal, but is not true otherwise. Answer. < x, y > is equal to 12. Answer. < x, y > is equal to +12 or -12. Comment. x and y are vectors, not scalars: saying that || x || = 3 does not mean that x is equal to +3 or -3, and similarly for y. Partially Correct Answer. < x, y > is between -12 and 12 inclusive. Comment. This is true for real inner product spaces, but for complex inner product spaces < x, y > can be complex. Answer. < x, y > is equal to +12, -12, +12i, or -12i. Correct Answer. < x, y > can be any complex number of magnitude 12 or less.