{
    "name": "Vector Dot Product Calculator",
    "url": "https://calculators.geekmidst.com/calculators/vector-dot-product-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/vector-dot-product-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Find the dot product of two vectors of any dimension, their magnitudes, the angle between them, and whether they are orthogonal or parallel.",
    "summary": "The dot product multiplies matching components and adds them. It also equals |a||b|cos θ, so it tells you the angle between the vectors.",
    "also_called": [
        "Scalar product calculator",
        "Inner product calculator",
        "Angle between vectors 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": "a",
            "label": "Vector a",
            "type": "text",
            "default": "1, 2, 3"
        },
        {
            "name": "b",
            "label": "Vector b",
            "type": "text",
            "default": "4, -5, 6"
        }
    ],
    "outputs": [
        {
            "key": "dot",
            "label": "a · b",
            "format": "number"
        },
        {
            "key": "mag_a",
            "label": "|a|",
            "format": "number"
        },
        {
            "key": "mag_b",
            "label": "|b|",
            "format": "number"
        },
        {
            "key": "cos",
            "label": "cos θ",
            "format": "text"
        },
        {
            "key": "angle",
            "label": "Angle θ between a and b",
            "format": "text"
        },
        {
            "key": "relation",
            "label": "Relationship",
            "format": "text"
        }
    ],
    "formula": {
        "text": "a · b = Σ aᵢbᵢ = |a||b| cos θ;  θ = 2·atan2(|â − b̂|, |â + b̂|)",
        "variables": {
            "aᵢ, bᵢ": "the components",
            "|a|, |b|": "the lengths",
            "θ": "the angle between the vectors",
            "â, b̂": "the unit vectors"
        }
    },
    "worked_example": {
        "inputs": {
            "a": "1, 2, 3",
            "b": "4, -5, 6"
        },
        "steps": [
            "a · b = 1·4 + 2·(−5) + 3·6 = 4 − 10 + 18 = 12.",
            "|a| = √14 ≈ 3.7417 and |b| = √77 ≈ 8.7750.",
            "cos θ = 12 ÷ (3.7417 × 8.7750) ≈ 0.36549, so θ ≈ 68.56°."
        ],
        "result": "(1, 2, 3) · (4, −5, 6) = 12, and the vectors are about 68.56° apart."
    },
    "interpretation": "A positive dot product means the angle is acute, a negative one obtuse, and zero means the vectors are perpendicular (orthogonal).",
    "assumptions": [
        "Both vectors are in the same Euclidean space with the standard dot product."
    ],
    "limitations": [
        "The angle is undefined if either vector is the zero vector."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "Why is the angle not just arccos of cos θ?",
            "answer": "arccos loses accuracy for nearly parallel vectors. The calculator uses the equivalent formula 2·atan2(|â − b̂|, |â + b̂|), which stays accurate at every angle."
        },
        {
            "question": "What is the dot product used for?",
            "answer": "Work done by a force (F · d), projections, testing perpendicularity, and similarity scores between vectors."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Vector Cross Product Calculator",
            "url": "https://calculators.geekmidst.com/calculators/vector-cross-product-calculator"
        },
        {
            "name": "Vector Projection Calculator",
            "url": "https://calculators.geekmidst.com/calculators/vector-projection-calculator"
        },
        {
            "name": "Distance Calculator",
            "url": "https://calculators.geekmidst.com/calculators/distance-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}