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Vectors can be added (tip-to-tail or component-wise), subtracted (add the negative), and scaled by a scalar (multiply each component). These operations are performed component-wise: (a1, b1) + (a2, b2) = (a1 + a2, b1 + b2) and c*(a, b) = (ca, cb). Linear combinations of vectors (c1*v1 + c2*v2) span the plane and are the foundation of linear algebra.
Practice both graphically (tip-to-tail addition, parallelogram rule) and algebraically (component-wise). Show that scalar multiplication stretches or shrinks and can reverse direction. Connect to physics: net force is vector addition, scaling a velocity vector changes speed.
From your study of vectors in two dimensions, you know that a vector represents both a magnitude and a direction, written as an ordered pair (a, b) in component form. The operations on vectors — addition, subtraction, and scalar multiplication — are all defined component-wise, which means you apply the operation separately to each coordinate. This design ensures that vector operations are consistent with the geometric interpretations you already have in mind.
Vector addition (a₁, b₁) + (a₂, b₂) = (a₁+a₂, b₁+b₂) has a precise geometric meaning: the tip-to-tail rule. Draw the first vector from the origin, then place the tail of the second at the tip of the first. The sum is the vector from the origin to the new tip. The parallelogram rule is equivalent — place both vectors tail-to-tail and complete the parallelogram; the diagonal is the sum. Either way, the result is the component-wise sum, and both geometric pictures are saying the same thing in different arrangements. In physics, this is how you add forces or velocities: a boat moving at (3, 0) m/s in a river with current (0, 2) m/s ends up moving at (3, 2) m/s.
Scalar multiplication c·(a, b) = (ca, cb) scales both components by the same factor. Geometrically, it stretches or shrinks the vector by |c| and reverses its direction if c is negative. The direction can only be preserved (c > 0) or reversed (c < 0) — scalar multiplication cannot rotate a vector to any arbitrary angle. That is a key constraint: to change direction to something other than 180°, you need vector addition. Vector subtraction u - v is defined as u + (-v), where -v = (-1)·v is the reversal. Geometrically, u - v is the vector pointing from the tip of v to the tip of u — the "displacement" from v to u.
Linear combinations c₁v₁ + c₂v₂ are the core of linear algebra. By varying c₁ and c₂ over all real numbers, you generate every vector in the plane (assuming v₁ and v₂ point in different directions). This is what "spanning the plane" means. If you think of v₁ = (1, 0) and v₂ = (0, 1), then c₁·(1,0) + c₂·(0,1) = (c₁, c₂) — any point in the plane is reachable. This idea extends directly to higher dimensions and to abstract vector spaces, making the three operations you are learning now the foundation of all of linear algebra.