The vectors $(1, 0)$ and $(0, 0)$ in $\reals^2$

Terms

Term	This example	Why
orthogonal	✓	The vectors $(1, 0)$ and $(0, 0)$ are orthogonal, because their dot product is $1 \cdot 0 + 0 \cdot 0 = 0$.
linearly independent	✗	The vectors $(1, 0)$ and $(0, 0)$ are not linearly independent, because $0 \cdot (1, 0) + 1 \cdot (0, 0) = (0, 0)$ is a linear combination with a nonzero coefficient that equals the zero vector.
perpendicular	✗	The vectors $(1, 0)$ and $(0, 0)$ are not perpendicular, because the zero vector has no direction, so the vectors form no angle.
pairwise linearly independent	✗	The vectors $(1, 0)$ and $(0, 0)$ are not pairwise linearly independent, because $(0, 0) = 0 \cdot (1, 0)$ is a scalar multiple of $(1, 0)$.
orthonormal	✗	The vectors $(1, 0)$ and $(0, 0)$ are not orthonormal, because $(0, 0)$ has length $0$.

Linearly independent vs. orthogonal: The vectors $(1, 0)$ and $(0, 0)$ show why orthogonal vectors must be nonzero to be linearly independent: the zero vector is orthogonal to every vector, but every list containing the zero vector is linearly dependent.
Linearly independent vs. pairwise non-parallel
Right angle vs. perpendicular vs. orthogonal vs. normal: The vectors $(1, 0)$ and $(0, 0)$ show that orthogonality is defined by the inner product alone, so the zero vector is orthogonal to every vector even though the zero vector forms no angle.