Final answer:
A vector x can be expressed as the sum of two vectors, with properties such as commutativity and associativity aiding this process. In a coordinate system, we can find the sum of vectors by adding their respective scalar components. The sum is known as the resultant vector, and it is formed by straightforward component-wise addition.
Step-by-step explanation:
The question seems to be about expressing a vector x as the sum of two other vectors, where one is in the span of a certain set and the other is in the orthogonal complement of that span. To do this, we need to utilize the properties of vector addition and scalar multiplication. For example, consider two vectors ℓ and ℓ. According to the commutative property of vector addition (ℓ + ℓ = ℓ + ℓ), the order of adding these vectors does not change the resultant vector. Similarly, the associative property allows us to group vectors together in an addition without affecting the sum, and distributive properties allow us to multiply vectors by scalars in a linear fashion.
When vectors are represented in a coordinate system, we can use their scalar components to find their sums. If vector A has components (Ax, Ay, Az) and vector B has components (Bx, By, Bz), then the resultant vector ℞ = A + B is given by ℞ = (Ax + Bx) Ģ + (Ay + By) ij + (Az + Bz) k. Resolving vectors into their scalar components helps us to perform vector addition analytically.
The resultant vector, also called the sum, can be described as ℞AD = ℞AC + ℞CD, which involves adding the vectors component by component.