Vectors and Matrices

Learning Objectives

Define a vector and distinguish it from a scalar.

Perform basic vector operations: addition, scalar multiplication, and the dot product.

Define a matrix and its dimensions.

Calculate the determinant of a 2x2 matrix.

Vectors: More than Just a Number

A scalar is a quantity that has only magnitude (a size or numerical value). Examples: speed, mass, temperature.

A vector is a quantity that has both magnitude and direction. Examples: velocity, force, acceleration.

Vectors are often represented as arrows or as ordered pairs/triplets of numbers, e.g., v = <3, 4>.

Vector Operations

Scalar Multiplication: Multiplying a vector by a scalar changes its magnitude (length). Multiplying <x, y> by a scalar c gives <cx, cy>.

Vector Addition: To add two vectors, add their corresponding components. <x₁, y₁> + <x₂, y₂> = <x₁ + x₂, y₁ + y₂>. Geometrically, this is represented by the 'head-to-tail' rule.

Dot Product: The dot product of two vectors is a scalar value. It is a measure of how much one vector points in the direction of the other.

Component form: <x₁, y₁> · <x₂, y₂> = x₁x₂ + y₁y₂.

An important property: The dot product of two perpendicular (orthogonal) vectors is zero.

Matrices: Arrays of Numbers

A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are given as rows x columns.

A 2x2 matrix looks like:

[ a b ]

[ c d ]

The Determinant

The determinant is a special scalar value that can be calculated from a square matrix. It has many important applications, such as in solving systems of linear equations and in linear algebra.

For a 2x2 matrix A = [ [a, b], [c, d] ], the determinant, denoted as det(A) or |A|, is calculated as:

det(A) = ad - bc

In words: multiply the main diagonal (top-left to bottom-right) and subtract the product of the other diagonal (top-right to bottom-left).

Example: Find the determinant of the matrix [ [4, 7], [2, 3] ].

det = (4)(3) - (7)(2) = 12 - 14 = -2.

Key Terms

Vector: A quantity having direction as well as magnitude, especially as determining the position of one point in space relative to another.
Scalar: A quantity having only magnitude, not direction.
Dot Product: An algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number.
Matrix: A rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
Determinant: A scalar value that is a function of the entries of a square matrix, used in linear algebra.