Practice question · Sort into groups
Sort each pair of vectors by whether the two are orthogonal.
Groups: Orthogonal · Not orthogonal
- (3, 1) and (1, 3)
- (2, 1, -1) and (1, -1, 1)
- (1, -1, 2) and (2, 4, 1)
- (1, 2) and (2, -1)
- (1, 1) and (1, 1)
- (1, 0, 1) and (0, 5, 0)
Hints
- Compute each inner product; orthogonality is exactly the value 0.
- For the last pair: 2 - 4 + 2.
Show the answer
Orthogonal: (1, 2) and (2, -1), (1, 0, 1) and (0, 5, 0), (2, 1, -1) and (1, -1, 1), (1, -1, 2) and (2, 4, 1)
Not orthogonal: (1, 1) and (1, 1), (3, 1) and (1, 3)
Why
The inner products are 0, 2, 0, 0, 6 and 0. Note the second pair: a nonzero vector is never orthogonal to itself, precisely because positive definiteness forces the inner product of a nonzero vector with itself to be positive.
Practise Inner Product Spaces and Orthogonality
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