{
  "id": "unit-tangent-vector",
  "version": "0f4f93444b64",
  "status": "published",
  "name": "Unit Tangent Vector Calculator",
  "question": "What is the unit tangent vector T(t)?",
  "summary": "Finds the unit tangent vector T(t) = r′(t)/‖r′(t)‖ of a curve r(t) in 2D or 3D, and its value at t = a.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/unit-tangent-vector-calculator",
  "markdown": "https://www.acalculator.org/math/unit-tangent-vector-calculator.md",
  "kind": "function",
  "method": "T(t) = r′(t) / ‖r′(t)‖, each component of r′ from a computer algebra system, checked numerically.",
  "assumptions": [
    "The variable is t; angles are in radians.",
    "T(a) needs r′(a) ≠ 0."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "x": {
        "title": "x(t)",
        "description": "The x component of r(t), a function of t.",
        "type": "string",
        "maxLength": 150
      },
      "y": {
        "title": "y(t)",
        "description": "The y component of r(t), a function of t.",
        "type": "string",
        "maxLength": 150
      },
      "z": {
        "title": "z(t) (empty for a plane curve)",
        "description": "The z component of r(t), a function of t.",
        "type": "string",
        "maxLength": 150
      },
      "a": {
        "title": "At t = a",
        "description": "Where to evaluate T: a number or a constant such as pi/2.",
        "type": "string",
        "maxLength": 40
      }
    }
  },
  "outputs": {
    "at": {
      "label": "T(a)",
      "description": "The unit tangent vector at t = a.",
      "format": "text"
    },
    "formula": {
      "label": "T(t) =",
      "description": "r′(t) divided by its length ‖r′(t)‖.",
      "format": "text"
    },
    "derivative": {
      "label": "r′(t) =",
      "description": "The tangent vector, each component differentiated.",
      "format": "text"
    },
    "speed": {
      "label": "‖r′(a)‖",
      "description": "The length of r′(a), the speed at t = a.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "x": "3t^2 + 2t",
      "y": "2 - 4t^3",
      "a": "1",
      "z": "6t + 5"
    },
    "outputs": {
      "at": "(4/sqrt(61), -6/sqrt(61), 3/sqrt(61))",
      "formula": "(6t + 2, -12t^2, 6)/sqrt(144t^4 + 36t^2 + 24t + 40)",
      "derivative": "(6t + 2, -12t^2, 6)",
      "speed": 15.620499351813308
    },
    "text": "The unit tangent vector at t = 1 is (4/sqrt(61), -6/sqrt(61), 3/sqrt(61))."
  },
  "examples": [
    {
      "given": {
        "x": "3t^2 + 2t",
        "y": "2 - 4t^3",
        "z": "6t + 5",
        "a": "1"
      },
      "expect": {
        "derivative": "(6t + 2, -12t^2, 6)",
        "formula": "(6t + 2, -12t^2, 6)/sqrt(144t^4 + 36t^2 + 24t + 40)",
        "at": "(4/sqrt(61), -6/sqrt(61), 3/sqrt(61))",
        "speed": 15.620499351813308
      },
      "source": "OpenStax, Calculus Volume 3, section 3.2 Calculus of Vector-Valued Functions, Example 3.7 (https://openstax.org/books/calculus-volume-3/pages/3-2-calculus-of-vector-valued-functions), part b: u′(t) = (6t + 2, −12t², 6); ‖u′(1)‖ = √244 = 2√61 by hand in content.mdx"
    },
    {
      "given": {
        "x": "cos(t)",
        "y": "sin(t)",
        "a": "pi/2"
      },
      "expect": {
        "derivative": "(-sin(t), cos(t))",
        "formula": "(-sin(t), cos(t))",
        "at": "(-1, 0)",
        "speed": 1
      },
      "source": "OpenStax, Calculus Volume 3, section 3.2 Calculus of Vector-Valued Functions, Example 3.7 (https://openstax.org/books/calculus-volume-3/pages/3-2-calculus-of-vector-valued-functions), part a: T(t) = −sin t i + cos t j; at t = π/2, T = (−1, 0)"
    },
    {
      "given": {
        "x": "t^2 - 3",
        "y": "2t + 1",
        "z": "t - 2",
        "a": "2"
      },
      "expect": {
        "at": "(4/sqrt(21), 2/sqrt(21), 1/sqrt(21))",
        "speed": 4.58257569495584
      },
      "source": "hand calculation in content.mdx: r′(2) = (4, 2, 1), ‖r′(2)‖ = √21; Python 3"
    }
  ],
  "sources": [
    "OpenStax, Calculus Volume 3, section 3.2 Calculus of Vector-Valued Functions (retrieved 2026-10-03): https://openstax.org/books/calculus-volume-3/pages/3-2-calculus-of-vector-valued-functions",
    "OpenStax, Calculus Volume 3, section 3.3 Arc Length and Curvature (retrieved 2026-10-03): https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature"
  ],
  "related": [
    "curvature",
    "unit-vector",
    "derivative",
    "magnitude"
  ],
  "changelog": []
}