Components describe the vector in a chosen system
A two-dimensional vector can describe a horizontal and vertical displacement. The coordinate system matters: the same underlying vector can have different components in a different basis.
Add and scale consistently
Add corresponding components to combine vectors. Multiplying by a scalar scales every component; a negative scalar also reverses direction. Sums of scaled vectors are linear combinations.
A mistake to avoid
Do not add a horizontal component to a vertical component just because they appear next to each other. Also distinguish a position point from a displacement vector when interpreting the result.
Worked example
Combine u = (2, −1) and v = (−3, 4)
Add matching components: u + v = (2 − 3, −1 + 4) = (−1, 3). Scaling u by 3 gives 3u = (6, −3). These calculations preserve the horizontal/vertical order.