What defines a vector in mathematics?

A vector in mathematics is fundamentally defined as a directed line segment. This means that a vector possesses both a magnitude (or length) and a direction. When visualizing a vector, you can think of an arrow: the length of the arrow indicates its magnitude, while the direction the arrow points represents the vector's direction. This duality of magnitude and direction is what differentiates vectors from other quantities.

For example, if you're considering movement, a vector can indicate both how far you need to go (magnitude) and in which direction to go (direction). This makes vectors essential in various fields, including physics, engineering, and computer graphics, where directionality is a crucial aspect of the quantities being modeled.

In contrast, other choices capture either only one of these properties or do not represent a vector accurately. For instance, a quantity involving magnitude only lacks direction, a point in space does not convey any directional information or magnitude, and a curved line cannot be categorized as a vector since it does not represent a single directional segment. Hence, the definition of a vector as a directed line segment encompasses both essential characteristics: magnitude and direction.