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2.1 Vector-Valued Functions and Space Curves

2 min readLast Updated on July 25, 2024

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 r(t)=f(t),g(t),h(t)\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle
  • Components include xx-component f(t)f(t), yy-component g(t)g(t), and zz-component h(t)h(t)
  • Parametric representation uses a parameter (usually tt) 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)x = f(t), y=g(t)y = g(t), z=h(t)z = h(t)
  • Vector form expresses space curves as r(t)=f(t)i+g(t)j+h(t)k\mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}
  • Common space curves include:
    • Helix: r(t)=cost,sint,t\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle spirals around z-axis
    • Circle in 3D space: r(t)=cost,sint,0\mathbf{r}(t) = \langle \cos t, \sin t, 0 \rangle traces unit circle in xy-plane

Graphing and Analysis

Graphs of vector-valued functions

Domain and range in vector functions

  • Domain considers interval of tt 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)

Term 1 of 16

Circle in 3D Space
See definition

A circle in 3D space is a set of points that are equidistant from a given point, called the center, and lies on a plane. This concept expands the traditional idea of a circle from a 2D plane to three dimensions, where the circle can be oriented in any direction and can have various positions relative to the coordinate axes. Understanding circles in 3D involves using vector-valued functions to describe their location and movement within that space.

Key Terms to Review (16)

Term 1 of 16

Circle in 3D Space
See definition

A circle in 3D space is a set of points that are equidistant from a given point, called the center, and lies on a plane. This concept expands the traditional idea of a circle from a 2D plane to three dimensions, where the circle can be oriented in any direction and can have various positions relative to the coordinate axes. Understanding circles in 3D involves using vector-valued functions to describe their location and movement within that space.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

Term 1 of 16

Circle in 3D Space
See definition

A circle in 3D space is a set of points that are equidistant from a given point, called the center, and lies on a plane. This concept expands the traditional idea of a circle from a 2D plane to three dimensions, where the circle can be oriented in any direction and can have various positions relative to the coordinate axes. Understanding circles in 3D involves using vector-valued functions to describe their location and movement within that space.



© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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