Unit 4 Vectors and Matrices: Directed Quantities, Parametric Motion, and Linear Maps

Vector

A quantity with both magnitude (size) and direction; unlike a scalar, which has only magnitude.

Scalar

A quantity with magnitude only (no direction), such as temperature.

Component form (2D vector)

A way to represent a 2D vector as ⟨a, b⟩, where a is the horizontal change and b is the vertical change.

Geometric representation of a vector

A vector drawn as an arrow whose length represents magnitude and whose orientation represents direction; it can be slid without changing the vector.

Column vector notation

Writing a 2D vector as a 2×1 column (e.g., [a; b]); equivalent to ⟨a, b⟩ and common in matrix contexts.

Negative vector

The vector with opposite direction, found by negating each component: if v=⟨a,b⟩ then −v=⟨−a,−b⟩.

Vector addition (component-wise)

If u=⟨ux,uy⟩ and v=⟨vx,vy⟩, then u+v=⟨ux+vx, uy+vy⟩.

Vector subtraction (component-wise)

If u=⟨ux,uy⟩ and v=⟨vx,vy⟩, then u−v=⟨ux−vx, uy−vy⟩ (i.e., add the opposite).

Scalar multiplication (of a vector)

Scaling a vector by k: if v=⟨vx,vy⟩ then kv=⟨kvx, kvy⟩; negative k also reverses direction.

Magnitude (length) of a 2D vector

For ⟨a,b⟩, the magnitude is |⟨a,b⟩|=√(a^2+b^2).

Unit vector

A vector with magnitude 1 that represents direction only.

Unit vector in the direction of ⟨a,b⟩

For nonzero ⟨a,b⟩, the unit vector is (1/√(a^2+b^2))⟨a,b⟩ (divide both components by the magnitude).

Direction angle (from +x-axis)

For a vector ⟨a,b⟩ with a≠0, the direction angle θ satisfies tan(θ)=b/a (and you must choose the correct quadrant).

Vector-valued function

A function that outputs a vector rather than a single number; in 2D it is typically r(t)=⟨x(t), y(t)⟩.

Position vector (in a vector-valued function)

The output r(t)=⟨x(t),y(t)⟩ interpreted as the position of a moving point at parameter value t (often time).

Parametric curve

The set of points (x(t), y(t)) traced as t varies over a domain; t controls where you are on the curve, not an axis on the graph.

Displacement from t=a to t=b

If r(t) is position, displacement over [a,b] is r(b)−r(a).

Average velocity (vector)

Displacement divided by elapsed time: (r(b)−r(a))/(b−a).

Eliminating the parameter

Rewriting a parametric curve as a single equation by solving for t and substituting, when possible (e.g., x=t into y=t^2−4 gives y=x^2−4).

Matrix

A rectangular array of numbers used to organize information and represent linear transformations (functions that take vectors to vectors).

Matrix dimension (size)

An m×n matrix has m rows and n columns.

Matrix multiplication (dimension rule)

If A is m×n and B is n×p, then AB is defined and has size m×p; multiplication is row-by-column (not entry-by-entry) and order matters.

Identity matrix (2×2)

I=[[1,0],[0,1]]; acts like “do nothing” for multiplication: I x = x and AI=IA=A (when dimensions match).

Matrix-vector multiplication (2D)

If A=[[a,b],[c,d]] and v=[x;y], then Av=[ax+by; cx+dy], giving the transformed vector.

Composition of linear transformations (matrices)

Applying B first then A corresponds to the product AB (order reverses relative to the wording).