What is a vector?
An arrow with a length and a direction — or, equivalently, a list of numbers that transforms the way everything else does.
Two views, same object
Geometrically, a vector in the plane is an arrow — it has a magnitude (how long) and a direction (which way). It doesn't have a position; if you slide it without rotating or stretching, it is still the same vector.
Algebraically, the same vector is just an ordered list of numbers: (3, −2) means "3 right, 2 down." The two views are equivalent — the arrow and the list carry the same information, and the right translation between them is what makes linear algebra work.
Operations
You can add vectors componentwise: (a, b) + (c, d) = (a + c, b + d). You can scale them by a number: k · (a, b) = (ka, kb). Together, addition and scaling are the two operations that make a vector space a vector space.
Geometrically, addition puts arrows tip-to-tail; scaling stretches or shrinks them, and flips direction if k is negative.
Worked example
Let u = (1, 2) and v = (3, −1). Compute 2u − v.
2u = (2, 4) 2u − v = (2 − 3, 4 − (−1)) = (−1, 5)The result is the vector (−1, 5) — one unit left, five units up. Notice how every step is componentwise; vectors never mix across their slots.
Flashcard · Unit 1
Multiply matching components, then sum.
Check your understanding
Given u = (2, −1) and v = (1, 3), what is u + v?