PEPCVF - Vectors and Vector-Valued Functions

Vectors and Vector-Valued Functions

Vectors in the Plane

Recall from precalculus and the Preparation for Calculus module, a vector is a quantity that has both magnitude and direction and is symbolized by boldface type (v) or by an arrow above the letter equation image indicator $LaTeX: \left(\vec{v}\,\right)$ $\left(\vec{v}\,\right)$ . Just as with polar coordinates, a vector in the plane can be represented in many different ways. Directed line segments representing a vector all point in the same direction and all are of the same length. The directed line segment whose initial point is the origin is said to be in standard position and is called the position vector. The presentation below may serve to refresh your memory about how to visualize vectors in the plane.

Component Form

If v is a vector in the plane whose initial point is the origin and whose the terminal point has coordinates (v₁, v₂), then the component form of v is given by <v₁, v₂ >. A more algebraic approach to performing operations on vectors uses the component method, which involves taking vectors, separating them into x and y coordinates, and adding or multiplying by a scalar the x and y coordinates separately.

Length of a Vector

The length or magnitude of a vector v = < v₁, v₂ > is | v | = equation image indicator $LaTeX: \sqrt[]{v^2_1+v^2_2}$ $\sqrt[]{v^2_1+v^2_2}$ . The length of v is also called the norm of v and is written as |v| or . The only vector with length 0 and no specific direction is the zero vector, which is denoted by 0 = < 0, 0 >. Geometrically, if both the initial point and the terminal point lie at the origin, then v is the zero vector.

Vector Operations and Their Properties

Two of the most frequently used vector operations are vector addition and scalar multiplication. Using component form, the sum of two vectors u = < u₁, u₂ > and v = < v₁, v₂ > is the vector u + v = < u₁ + v₁, u₂ + v₂ >. Recall that a scalar is a real number and can be positive, negative, or zero. Scalars behave like scaling factors when applied to vectors. When c is a scalar, the scalar multiple of c and v is the vector cv = < cv₁, cv₂ >. It follows that the negative of v is a vector -v = (-1)v = < -v₁, -v₂ > and the difference of u and v is u - v = < u₁ - v₁, u₂ - v₂ >.

Another useful vector operation is the dot product. If u = < u₁, u₂ > and v = < v₁, v₂ >, then the dot product of u and v is a scalar (number) given by u · v = u₁v₁ + u₂v₂. When the resulting numerical value is positive, the angle between the vectors is acute, and when the value is negative, the angle is obtuse. If the dot product between two nonzero vectors is zero, the vectors are perpendicular or orthogonal.

Geometrically, the dot product provides a means for determining the angle between two nonzero vectors u and v whose representations start at the origin, where equation image indicator $LaTeX: 0\le\theta\le\pi$ $0\le\theta\le\pi$ as follows:

equation image indicator $LaTeX: \cos\theta=\frac{\textbf{a}\:\cdot \textbf{b}}{\left|\textbf{a}\right|\left|\textbf{b}\right|}$ $\cos\theta=\frac{\textbf{a}\:\cdot \textbf{b}}{\left|\textbf{a}\right|\left|\textbf{b}\right|}$

The next presentation reviews vector operations and provides illustrative examples.

Properties of vector operations are summarized below.

Property	Mathematical Interpretation
Commutative	u + v = v + u
Associative	(u + v ) + w = u + (v + w)
Additive Identity	u + 0 = u
Additive Inverse	u + (- u) = 0
Associative Property for Scalars	c(du) = (cd)u
Distributive	(c + d)u = cu + du
Distributive	c(u + v) = cu + cv
Multiplicative Identity	1(u) = u
Zero Property of Multiplication	0(u) = 0

Standard Unit Vectors

It is often useful to find a unit vector that has the same direction as a given vector. Any vector u whose length is 1 is called a unit vector. The standard unit vectors in the plane are < 1, 0 > and < 0, 1 > and are denoted by i = < 1, 0 > and j = < 0, 1 >. The vector v = v₁ i + v₂ j is called a linear combination of i and j. The scalar v₁ is called the i-component of the vector v and v₂ is the j-component. If u is the unit vector obtained by rotating i through an angle equation image indicator in the positive direction (counterclockwise) from the positive x-axis to u, then u has a horizontal component cos and a vertical component sin and is expressed as u = (cos ) i + (sin ) j. It follows that in general, if v ≠ 0, then the unit vector that has the same direction as v is equation image indicator $LaTeX: \frac{\vec{v}\,}{\left|\vec{v}\,\right|}$ $\frac{\vec{v}\,}{\left|\vec{v}\,\right|}$ .

The presentation below focuses on determining a unit vector in the same direction as a given vector and on determining a vector that intersects the unit circle.

The next two presentations address unit vector notation and expressing a vector as the scaled sum of unit vectors.

A summary and examples of the introductory vector concepts reviewed above and studied in precalculus is the focus of the video below.

Vector-Valued Functions

A vector-valued function is a function of the form r(t) = f(t) i + g(t) j in the plane or r(t) = f(t) i + g(t) j + h(t) k in space, where the component functions f, g, and h are real-valued functions of the parameter t. Vector-valued functions may also be denoted by r(t) = < f(t), g(t) > or r(t) = < f(t), g(t), h(t) >. The domain of a vector-valued function is a set of real numbers determined by the intersection of the domains of the component functions f, g, and h and the range is a set of vectors. An example of finding the domain of a vector-valued function is presented below.

The limit of a vector-valued function is defined similarly to how a real-valued function is defined. If r is a vector-valued function such that r(t) = f(t) i + g(t) j in the plane or r(t) = f(t) i + g(t) j + h(t) k in space, then equation image indicator $LaTeX: \lim_{t \to a}\textbf{r}(t)=[\lim_{t \to a}f(t)]\textbf{i}+[\lim_{t \to a}g(t)]\textbf{j}\:or\:\lim_{t \to a}\textbf{r}(t)=[\lim_{t \to a}f(t)]\textbf{i}+[\lim_{t \to a}g(t)]\textbf{j}+[\lim_{t \to a}f(t)]\textbf{k}$ $\lim_{t \to a}\textbf{r}(t)=[\lim_{t \to a}f(t)]\textbf{i}+[\lim_{t \to a}g(t)]\textbf{j}\:or\:\lim_{t \to a}\textbf{r}(t)=[\lim_{t \to a}f(t)]\textbf{i}+[\lim_{t \to a}g(t)]\textbf{j}+[\lim_{t \to a}f(t)]\textbf{k}$ provided f, g, and h have limits as equation image indicator . The equivalent representation in component form is $LaTeX: \lim_{t \to a}\textbf{r}(t)=<\lim_{t \to a}f(t),\lim_{t \to a}g(t)>$ $\lim_{t \to a}\textbf{r}(t)=<\lim_{t \to a}f(t),\lim_{t \to a}g(t)>$ in the plane or $LaTeX: \lim_{t \to a}\textbf{r}(t)=<\lim_{t \to a}f(t),\lim_{t \to a}g(t),\lim_{t \to a}h(t)>$ $\lim_{t \to a}\textbf{r}(t)=<\lim_{t \to a}f(t),\lim_{t \to a}g(t),\lim_{t \to a}h(t)>$ in space. In short, the limit of a vector function is accomplished by taking the limits of its component functions. Note that the orientation of the curve r(t) can be used to define one-sided limits of vector-valued functions. The next presentation illustrates this limit definition is applied to several vector-valued functions.

A vector-valued function r is continuous at a if the limit of r(t) exists as t approaches a and equation image indicator $LaTeX: \lim_{t \to a}\textbf{r}(t)=\textbf{r}(a)$ $\lim_{t \to a}\textbf{r}(t)=\textbf{r}(a)$ . It is a continuous function if it is continuous at every point in its domain.

Planar Curves Defined by Vector-Valued Functions

If r(t) is represented by a position vector for each number t in the domain, then as t takes on all values in the domain, the terminal point (f(t), g(t)) of the position vector traces out the plane curve whose parametric representation is x = f(t), y = g(t). When t represents time, a vector-valued function represents motion along a curve. More generally, a vector-valued function can be used to trace the graph of a curve. In both situations the terminal point of the position vector r(t) coincides with the point (x, y) on the curve defined by the parametric equations. The curve's orientation is indicated by an arrowhead on the curve pointing in the direction of increasing values of t.

It is often helpful to be able to transform a static rectangular equation into a vector-valued function. The presentation below illustrates how this is accomplished.

Vector-valued functions and their graphs in both two dimensions and three dimensions and rectangular equations that correspond to vector-valued functions are the focus of the next illustration.

Vectors and Vector-Valued Functions Practice

Vectors and Vector-Valued Functions: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of vectors and vector-valued functions.

IMAGES CREATED BY GAVS