{
    "name": "Vector Magnitude Calculator",
    "url": "https://calculators.geekmidst.com/calculators/vector-magnitude-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/vector-magnitude-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Work out the magnitude (Euclidean length) of a vector from its components, plus its squared length, taxicab (L1) norm and maximum norm.",
    "summary": "A vector’s magnitude is the square root of the sum of its squared components, which is Pythagoras’ theorem in as many dimensions as you need.",
    "also_called": [
        "Vector length calculator",
        "Vector norm 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": "2, 3, 6"
        }
    ],
    "outputs": [
        {
            "key": "magnitude",
            "label": "Magnitude |v|",
            "format": "number"
        },
        {
            "key": "magnitude_sq",
            "label": "Squared magnitude |v|²",
            "format": "number"
        },
        {
            "key": "l1",
            "label": "Taxicab norm (L1)",
            "format": "number"
        },
        {
            "key": "linf",
            "label": "Maximum norm (L∞)",
            "format": "number"
        },
        {
            "key": "dimension",
            "label": "Number of components",
            "format": "integer"
        }
    ],
    "formula": {
        "text": "|v| = √(v₁² + v₂² + … + vₙ²);  L1 = Σ|vᵢ|;  L∞ = max|vᵢ|",
        "variables": {
            "vᵢ": "the components of the vector",
            "n": "the number of components"
        }
    },
    "worked_example": {
        "inputs": {
            "v": "2, 3, 6"
        },
        "steps": [
            "Square the components: 4, 9 and 36.",
            "Add them: 4 + 9 + 36 = 49.",
            "Take the square root: √49 = 7."
        ],
        "result": "The vector (2, 3, 6) has a magnitude of exactly 7."
    },
    "interpretation": "The magnitude is the straight-line length of the arrow. The L1 norm is the distance travelled along the axes (city blocks), and the L∞ norm is the largest single component.",
    "assumptions": [
        "Components are in an ordinary Cartesian (orthonormal) basis."
    ],
    "limitations": [
        "For physical vectors, all components must be in the same unit for the length to make sense."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "Is magnitude the same as distance?",
            "answer": "The magnitude of B − A is the distance between points A and B. Use the distance calculator to enter two points directly."
        },
        {
            "question": "Can a magnitude be negative?",
            "answer": "No. It is a length, so it is zero for the zero vector and positive otherwise."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Unit Vector Calculator",
            "url": "https://calculators.geekmidst.com/calculators/unit-vector-calculator"
        },
        {
            "name": "Vector Angle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/vector-angle-calculator"
        },
        {
            "name": "Distance Calculator",
            "url": "https://calculators.geekmidst.com/calculators/distance-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}