Final answer:
To be a vector space, a set must satisfy ten axioms, such as commutative addition and an additive identity. Vectors in a plane have magnitude and direction determined by their x and y components, altered by scalar multiplication. Addition in one dimension is simply combining magnitudes, while in a plane, the 'tip-to-tail' geometric method is used.
Step-by-step explanation:
To determine whether a set, together with the standard operations, is a vector space, it must satisfy ten specific axioms. These axioms include properties like vector addition being commutative, the existence of an additive identity (null vector), and the distributive properties of scalar multiplication over vector addition.
A vector can be represented in a plane by its components along the x and y axes. For any vector A, with horizontal component Ax and vertical component Ay, the magnitude of A can be found using the Pythagorean theorem as the square root of (Ax)^2 + (Ay)^2. The direction of the vector is typically given by the angle it makes with the horizontal axis.
Multiplying a vector by a scalar changes the magnitude of the vector without altering its direction, unless the scalar is negative, in which case the vector reverses direction. Vector addition in one dimension is straightforward, where vectors simply add or subtract their magnitudes depending on their direction. In a plane, vector addition can be performed geometrically by placing the tail of one vector at the head of the other and then drawing a vector from the tail of the first to the head of the second. This operation is known as the 'tip-to-tail' method.