{
    "name": "Unit Vector Calculator",
    "url": "https://calculators.geekmidst.com/calculators/unit-vector-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/unit-vector-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Normalise a vector by dividing each component by its magnitude to get the unit vector in the same direction, with the 2D direction angle.",
    "summary": "A unit vector keeps a vector’s direction but has length 1. Divide every component by the vector’s magnitude.",
    "also_called": [
        "Vector normalisation calculator",
        "Normalize vector calculator"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "v",
            "label": "Vector",
            "type": "text",
            "default": "3, 4"
        }
    ],
    "outputs": [
        {
            "key": "unit_vector",
            "label": "Unit vector û",
            "format": "text"
        },
        {
            "key": "magnitude",
            "label": "Original magnitude |v|",
            "format": "number"
        },
        {
            "key": "direction_2d",
            "label": "Direction angle (2D only)",
            "format": "text"
        },
        {
            "key": "dimension",
            "label": "Number of components",
            "format": "integer"
        }
    ],
    "formula": {
        "text": "û = v ÷ |v| = (v₁/|v|, v₂/|v|, …, vₙ/|v|);  |v| = √(Σ vᵢ²)",
        "variables": {
            "v": "the vector",
            "|v|": "its magnitude",
            "û": "the unit vector"
        }
    },
    "worked_example": {
        "inputs": {
            "v": "3, 4"
        },
        "steps": [
            "|v| = √(3² + 4²) = √25 = 5.",
            "Divide each component by 5: 3 ÷ 5 = 0.6 and 4 ÷ 5 = 0.8.",
            "Check: 0.6² + 0.8² = 0.36 + 0.64 = 1."
        ],
        "result": "The unit vector in the direction of (3, 4) is (0.6, 0.8)."
    },
    "interpretation": "The components of a unit vector are the direction cosines: the cosines of the angles it makes with each axis. Multiply the unit vector by any length to get a vector of that length in the same direction.",
    "assumptions": [
        "Components are in an ordinary Cartesian basis."
    ],
    "limitations": [
        "The zero vector has no direction, so it cannot be normalised.",
        "Components are shown to 6 decimal places, so their squares may sum to 1 only after rounding."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "How do I get a vector of length 10 in the same direction?",
            "answer": "Multiply each component of the unit vector by 10."
        },
        {
            "question": "Is the unit vector unique?",
            "answer": "For a given direction, yes. The opposite direction gives −û."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Vector Magnitude Calculator",
            "url": "https://calculators.geekmidst.com/calculators/vector-magnitude-calculator"
        },
        {
            "name": "Vector Angle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/vector-angle-calculator"
        },
        {
            "name": "Slope Calculator",
            "url": "https://calculators.geekmidst.com/calculators/slope-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}