2.1 Vector-Valued Functions and Space Curves

Vector-valued functions take us beyond single values, outputting vectors instead. They're like a GPS for math, giving us x, y, and z coordinates all at once. These functions help us map out paths in 3D space, tracing curves like helixes and circles.

Graphing these functions is like connecting the dots in three dimensions. We plot points, figure out which way the curve's going, and join them up. The domain tells us what inputs work, while the range shows all possible outputs. It's like setting the boundaries for our mathematical playground.

Vector-Valued Functions

Concept of vector-valued functions

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• Functions output vectors instead of scalar values, typically denoted as $\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle$
• Components include $x$-component $f(t)$, $y$-component $g(t)$, and $z$-component $h(t)$
• Parametric representation uses a parameter (usually $t$) to define each component
• Each component functions as a scalar-valued function, mapping parameter to real number

Representation of space curves

• Space curves trace paths in three-dimensional space using vector-valued functions
• Parametric equations define space curves: $x = f(t)$, $y = g(t)$, $z = h(t)$
• Vector form expresses space curves as $\mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}$
• Common space curves include:
  • Helix: $\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle$ spirals around z-axis
  • Circle in 3D space: $\mathbf{r}(t) = \langle \cos t, \sin t, 0 \rangle$ traces unit circle in xy-plane

Graphing and Analysis

Graphs of vector-valued functions

• Sketching techniques:
  1. Plot individual points for different $t$ values
  2. Identify curve's orientation and direction
  3. Connect points to form continuous curve
• Special cases:
  • Planar curves confined to single plane (ellipse)
  • Closed curves return to starting point (circle)
• Visualization tools:
  • Graphing calculators plot 3D curves
  • Computer software (Mathematica, MATLAB) generates interactive 3D models

Domain and range in vector functions

• Domain considers interval of $t$ where all component functions are defined
• Restrictions based on physical context limit domain (time cannot be negative)
• Range analysis determines set of all possible output vectors
• Geometric interpretation of range as set of points traced by curve
• Finding domain and range:
  • Examine each component function separately
  • Consider intersections of individual domains
• Special cases affect range:
  • Periodic functions create repeating patterns (sine wave)
  • Bounded ranges have limits (unit circle)
  • Unbounded ranges extend infinitely (spiral)