Vectors in the Plane: Coordinates, Length and Linear Independence

A vector in the plane is a quantity with both a length and a direction, written in coordinates as a = (a₁, a₂); its length is |a| = √(a₁² + a₂²), and two vectors are linearly independent when neither is a scalar multiple of the other. In brief: the vector (3, 4) has length √(9 + 16) = 5, the sum (3, 1) + (1, 2) is (4, 3), and (2, 6) and (1, 3) are linearly dependent because (2, 6) = 2 · (1, 3).

Vectors describe displacements, velocities and forces, and they are the basic language of analytic geometry. This guide explains vectors given by coordinates, addition and scalar multiplication, length and unit vectors, the vector between two points, and linear independence with the determinant test, with worked examples and a table of operations.

What is a vector in the plane?

A vector can be pictured as an arrow. Two arrows with the same length and direction represent the same vector, wherever they start. In a coordinate system, a vector is determined by its two components a = (a₁, a₂): the horizontal and vertical displacement. The zero vector (0, 0) has no direction. A point P = (x, y) corresponds to its position vector from the origin.

How do you add vectors and multiply by a number?

Addition works component by component: (a₁, a₂) + (b₁, b₂) = (a₁ + b₁, a₂ + b₂). Geometrically, place the start of b at the end of a (the triangle rule) or complete a parallelogram (the parallelogram rule). Multiplying by a number k scales each component: k(a₁, a₂) = (ka₁, ka₂). A negative k reverses the direction. Subtraction a − b = a + (−b).

Example 1: sum and difference

For a = (2, −1) and b = (−3, 4): a + b = (−1, 3), a − b = (5, −5), 3a = (6, −3), 2a − b = (4, −2) − (−3, 4) = (7, −6).

Example 2: solving a vector equation

Find x such that 2x + (1, 3) = (7, 1). Then 2x = (6, −2), so x = (3, −1).

How do you find the length of a vector?

By the Pythagorean theorem, |a| = √(a₁² + a₂²). The properties: |ka| = |k||a| and the triangle inequality |a + b| ≤ |a| + |b|. A unit vector has length 1. To normalize a nonzero vector, divide it by its length: â = a/|a|.

Example 3: length and unit vector

a = (5, −12): |a| = √(25 + 144) = 13. The unit vector in the same direction is (5/13, −12/13).

Example 4: a vector with a given length

A vector of length 10 in the direction of (3, 4): the unit vector is (3/5, 4/5), so the result is (6, 8).

How do you find the vector between two points?

For A = (x_A, y_A) and B = (x_B, y_B), the vector from A to B is AB = (x_B − x_A, y_B − y_A) (end minus start). Its length is the distance between the points. The midpoint of AB has the position vector (A + B)/2.

Example 5: a vector and the distance

A = (1, 2), B = (7, 10): AB = (6, 8), |AB| = 10, and the midpoint is (4, 6).

Example 6: the fourth vertex of a parallelogram

Three vertices of the parallelogram ABCD are A = (0, 0), B = (4, 1), C = (6, 5). Since AD = BC = (2, 4), D = (2, 4). Check: the diagonals AC and BD share the midpoint (3, 2.5) ✓.

When are two vectors linearly independent?

Vectors a and b are linearly dependent if one is a multiple of the other, equivalently if there are numbers α, β, not both 0, with αa + βb = 0. Otherwise they are independent. In the plane, a = (a₁, a₂) and b = (b₁, b₂) are independent if and only if the determinant a₁b₂ − a₂b₁ ≠ 0. Geometrically, dependent vectors are parallel and the parallelogram they span is flat; the determinant gives its signed area.

Two independent vectors form a basis: every vector c in the plane is a unique combination c = αa + βb. Three or more vectors in the plane are always dependent.

Example 7: testing independence

a = (2, 6), b = (1, 3): 2 · 3 − 6 · 1 = 0, so they are dependent (a = 2b). a = (1, 2), b = (3, 4): 1 · 4 − 2 · 3 = −2 ≠ 0, so they are independent.

Example 8: coordinates in a new basis

Write c = (7, 4) as αa + βb with a = (1, 2) and b = (3, 4); the system is α + 3β = 7, 2α + 4β = 4. From the first, α = 7 − 3β; substituting gives 14 − 6β + 4β = 4, so β = 5 and α = −8. Check: −8(1, 2) + 5(3, 4) = (7, 4) ✓.

Example 9: a parameter

For which k are (k, 4) and (1, k) dependent? k · k − 4 · 1 = 0, so k = 2 or k = −2.

Table of operations

Where does the vector concept come from?

The parallelogram rule for composing forces was known by the sixteenth and seventeenth centuries (it appears in the works of Simon Stevin and in Newton’s Principia). The modern theory developed in the nineteenth century: William Rowan Hamilton introduced the word “vector” in connection with quaternions in the 1840s, Hermann Grassmann developed a general theory of linear spaces, and Josiah Willard Gibbs and Oliver Heaviside shaped the vector notation used today in the 1880s. The complex numbers of the complex plane give another way of describing plane vectors.

Where are vectors used?

Vectors describe forces and velocities in physics, moves of characters and cameras in computer graphics, and routes and wind drift in navigation. In data science, a vector of numbers represents a data point. The tools of this article are extended, which measures angles and projections.

Which mistakes are common?

Reversing start and end

AB = B − A. Computing A − B gives the opposite vector.

Adding lengths instead of vectors

|a + b| is generally not |a| + |b|; equality holds only for parallel vectors with the same direction. For a = (3, 0), b = (0, 4): |a + b| = 5, not 7.

Confusing a point with a vector

Points cannot be added, but their position vectors can. The sum of two points is meaningless; the midpoint is (A + B)/2.

Forgetting to normalize

A unit vector must have length exactly 1. Always divide by |a| and check the result.