Vectors describe quantities with both size and direction, and they are also a powerful way to locate and move between points. In algebra and geometry, a vector can be scaled, added, and used as a building block for other vectors. Linear combinations show how several vectors can work together to reach a target point.
This idea matters because it connects coordinate geometry, systems of equations, transformations, and many applications in physics and computer graphics.
A linear combination is made by multiplying vectors by scalars and then adding the results. In 2D, two nonparallel vectors can span the entire plane, meaning every point can be reached by some combination of them. If the vectors point along the usual coordinate axes, their scalar weights are just the x and y coordinates.
If the vectors are tilted, the same target point may still be reachable, but the needed weights come from solving equations.
Key Facts
- A vector in 2D can be written as v = <x, y>.
- Scalar multiplication changes magnitude and may reverse direction: c<x, y> = <cx, cy>.
- Vector addition combines components: <a, b> + <c, d> = <a + c, b + d>.
- A linear combination has the form c1v1 + c2v2 + ... + cnvn.
- The span of vectors is the set of all possible linear combinations of those vectors.
- Two 2D vectors u = <a, b> and v = <c, d> span the plane if ad - bc is not 0.
Vocabulary
- Vector
- A vector is a quantity or object with magnitude and direction, often represented by an ordered list of components.
- Scalar
- A scalar is a single number that can stretch, shrink, or reverse a vector when multiplied by it.
- Linear combination
- A linear combination is a sum of vectors after each vector has been multiplied by a scalar.
- Span
- The span of a set of vectors is the collection of all vectors that can be made from their linear combinations.
- Basis
- A basis is a set of vectors that spans a space and uses no redundant vectors.
Common Mistakes to Avoid
- Adding vector magnitudes instead of components is wrong because direction matters. Add x-components together and y-components together.
- Thinking scalar multiplication only changes length is wrong because a negative scalar also reverses the vector direction. For example, -2v points opposite v and has twice the magnitude.
- Assuming any two vectors span the plane is wrong because parallel vectors only reach points on one line. In 2D, the determinant ad - bc must be nonzero.
- Confusing a point with a vector can lead to incorrect diagrams. A point is a location, while a vector describes a displacement that can be drawn from many starting positions.
Practice Questions
- 1 Let u = <2, 1> and v = <1, 3>. Compute 2u - v.
- 2 Find scalars a and b such that a<1, 2> + b<3, 1> = <7, 8>.
- 3 Vectors p = <2, 4> and q = <1, 2> are used to make linear combinations. Explain whether they span the entire plane and justify your answer.