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Vectors are added and subtracted component-wise: (u₁ + v₁, u₂ + v₂, ..., uₙ + vₙ). Geometrically, addition follows the parallelogram rule; subtraction finds the vector between two points. These operations satisfy closure, associativity, and commutativity, forming the foundation of vector space structure.
Visualize in R2 using arrows. Add vectors tip-to-tail or using the parallelogram method. Verify algebraically with components. Then practice with higher dimensions using notation only.
You already know that a vector in Rⁿ is an ordered list of n real numbers — a point, or equivalently an arrow, in n-dimensional space. Vector addition is the simplest thing you can do with two such arrows: add them entry by entry. If u = (u₁, u₂) and v = (v₁, v₂), then u + v = (u₁ + v₁, u₂ + v₂). The component-wise rule is purely algebraic, but geometry makes it vivid.
In the plane, picture u as an arrow from the origin to some point, and v as another. The tip-to-tail rule says: to add them, place the tail of v at the tip of u. The result is the arrow from the origin to the new tip. Equivalently, u + v is the diagonal of the parallelogram whose sides are u and v. These two geometric pictures — tip-to-tail and the parallelogram — are equivalent, and either one lets you visualize addition in R² or R³ before generalizing to higher dimensions where pictures are no longer available.
Vector subtraction u − v = u + (−v) is just addition of the negated vector. Geometrically, −v is the same arrow pointing in the opposite direction. A particularly useful interpretation: u − v is the vector *from* v *to* u. If u and v are position vectors of two points P and Q, then u − v is the displacement from Q to P. This shows up constantly in geometry — the vector connecting two points is always a difference of their position vectors.
One thing to watch: adding magnitudes is wrong. If |u| = 3 and |v| = 4, then |u + v| is *not* 7 in general. It equals 7 only if the vectors point in exactly the same direction. If they are perpendicular, it equals 5 (by the Pythagorean theorem), and if they are antiparallel, it equals 1. This is why the triangle inequality ||u + v|| ≤ ||u|| + ||v|| holds with equality only in the collinear case. The component-wise addition rule is always exact; the magnitude has to be computed from the sum, not from the summands' magnitudes.