{
    "name": "Vector Angle Calculator",
    "url": "https://calculators.geekmidst.com/calculators/vector-angle-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/vector-angle-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Find the angle between two vectors in degrees and radians from their dot product and magnitudes, and see whether they are perpendicular or parallel.",
    "summary": "The angle between two vectors comes from the dot product: cos θ = (A·B) ÷ (|A| |B|). Enter the components separated by commas, in any number of dimensions.",
    "also_called": [
        "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": "3, 4"
        },
        {
            "name": "b",
            "label": "Vector B",
            "type": "text",
            "default": "4, 3"
        }
    ],
    "outputs": [
        {
            "key": "angle_deg",
            "label": "Angle between the vectors",
            "format": "number"
        },
        {
            "key": "angle_rad",
            "label": "Angle in radians",
            "format": "number"
        },
        {
            "key": "cos_theta",
            "label": "cos θ",
            "format": "number"
        },
        {
            "key": "dot",
            "label": "Dot product A·B",
            "format": "number"
        },
        {
            "key": "mag_a",
            "label": "|A|",
            "format": "number"
        },
        {
            "key": "mag_b",
            "label": "|B|",
            "format": "number"
        },
        {
            "key": "relation",
            "label": "Relationship",
            "format": "text"
        }
    ],
    "formula": {
        "text": "cos θ = (A·B) ÷ (|A| × |B|);  A·B = Σ aᵢbᵢ;  |A| = √(Σ aᵢ²);  θ = arccos(cos θ)",
        "variables": {
            "A, B": "the two vectors",
            "aᵢ, bᵢ": "their components",
            "θ": "the angle between them, from 0° to 180°"
        }
    },
    "worked_example": {
        "inputs": {
            "a": "3, 4",
            "b": "4, 3"
        },
        "steps": [
            "A·B = 3×4 + 4×3 = 24.",
            "|A| = √(9+16) = 5 and |B| = 5.",
            "cos θ = 24 ÷ 25 = 0.96, so θ = arccos 0.96 ≈ 16.2602°."
        ],
        "result": "The vectors (3, 4) and (4, 3) are about 16.26° apart."
    },
    "interpretation": "A dot product of zero means a right angle; a positive one means less than 90° and a negative one more than 90°. The angle never exceeds 180° because it is measured the short way round.",
    "assumptions": [
        "Both vectors are in the same space and use the same orthonormal (ordinary Cartesian) basis."
    ],
    "limitations": [
        "The angle is undefined if either vector is zero.",
        "This is the unsigned angle; a signed (clockwise or anticlockwise) angle needs a 2D orientation convention."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "Does the order of the vectors matter?",
            "answer": "No. The dot product is symmetric, so the angle from A to B equals the angle from B to A."
        },
        {
            "question": "Why is cos θ clamped?",
            "answer": "Rounding can push the ratio a hair above 1 for parallel vectors; it is limited to the range −1 to 1 so arccos stays defined."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Vector Magnitude Calculator",
            "url": "https://calculators.geekmidst.com/calculators/vector-magnitude-calculator"
        },
        {
            "name": "Unit Vector Calculator",
            "url": "https://calculators.geekmidst.com/calculators/unit-vector-calculator"
        },
        {
            "name": "Triangle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/triangle-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}