Final answer:
The statement that the shortest vector in an orthogonal lattice basis is the shortest column is false. An orthogonal lattice basis comprises vectors that are perpendicular to each other, and the shortest vector is the one with the smallest length, not necessarily the shortest column.
Step-by-step explanation:
False. When referring to an orthogonal lattice basis, the shortest vector is not necessarily the shortest column. An orthogonal basis consists of vectors that are all perpendicular to each other, but their lengths can vary. The length of a vector is the distance from the origin to its endpoint, and in an orthogonal basis, the shortest vector is the one with the smallest length regardless of its position as a row or column.
Vector components can indeed form the shape of a right angle triangle, and you can use the Pythagorean theorem to calculate the length of the resultant vector when two vectors are at right angles. However, every 2-D vector cannot be expressed simply as the product of its x and y-components; they can be represented by a sum of the components multiplied by the corresponding unit vectors in each direction.
Beth's comment about columns being closer to each other at the edges suggests non-uniformity, but without additional context, it does not directly relate to orthogonal lattices or vector components.