如何找到向量之间的角度？

让我们采用一些“参考”（“起始”）向量，我们将其视为原点。我采用了向量 (1; 0) - 这会稍微简化计算。
代码将在 C# 中：

struct Vector
{
    public double X { get; }
    public double Y { get; }

    public Vector(double x, double y)
    {
        X = x; Y = y;
    }

    public static readonly Vector Reference = new Vector(1, 0);

    // Угол в градусах относительно опорного, возвращает значение из диапазона (-180°; 180°]
    public static double AngleOfReference(Vector v)
        => Math.Atan2(v.Y, v.X) / Math.PI * 180;
}

然后，有序向量对之间的角度将计算为： Vector.AngleOfReference(a) - Vector.AngleOfReference(b);，但是，正如@VladD 在评论中正确建议的那样，输出可以是 (-360; 360] 范围内的值，它需要“归一化”，所以我仍然会在结构中添加一个方法来计算角度：

public static double AngleOfVectors(Vector first, Vector second)
{
    var angle = AngleOfReference(first) - AngleOfReference(second);
    if (angle > 180) angle -= 360;
    if (angle <= -180) angle += 360;
    return angle;
}

这将为我们提供一个恰好在 (-180°; 180°] 范围内的角度。

另一个小改进，我将给出结果结构的完整代码：

struct Vector
{
    public double X { get; }
    public double Y { get; }

    public Vector(double x, double y)
    {
        X = x; Y = y;
    }

    public static readonly Vector Reference = new Vector(1, 0);

    public static double AngleOfReference(Vector v)
        => NormalizeAngle(Math.Atan2(v.Y, v.X) / Math.PI * 180);

    public static double AngleOfVectors(Vector first, Vector second)
        => NormalizeAngle(AngleOfReference(first) - AngleOfReference(second));

    private static double NormalizeAngle(double angle)
    {
        bool CheckBottom(double a) => a >= 0;
        bool CheckTop(double a) => a < 360;

        double turn = CheckBottom(angle) ? -360 : 360;
        while (!(CheckBottom(angle) && CheckTop(angle))) angle += turn;
        return angle;
    }
}

通过调整NormalizeAngle您可以选择所需的输出范围。在上面的代码[0°; 360°)中，如果您需要它(-180°; 180°]，请像这样更正它：

bool CheckBottom(double a) => a > -180;
bool CheckTop(double a) => a <= 180;

基本的一切都很简单！

计算向量之间的值：

// создаем первый вектор
var start1 = new Point { X = 1, Y = 1 };
var end1 = new Point { X = 2, Y = 2 };
var vector1 = Vector.FromPoints(start1, end1);

// создаем второй вектор
var start2 = new Point { X = 2, Y = 2 };
var end2 = new Point { X = 1, Y = 1 };
var vector2 = Vector.FromPoints(start2, end2);

// вычисляем угол в радианах
var angleRad = Vector.CalculateAngleBetween(vector1, vector2);
// переводим угол в градусы
var angleDegree = angleRad / Math.PI * 180;

Console.WriteLine(angleDegree);

两个类Point和Vector：

public class Point
{
    public double X { get; set; }

    public double Y { get; set; }
}

public class Vector
{
    public double X { get; private set; }

    public double Y { get; private set; }

    public double Length { get; private set; }


    private Vector(double x, double y)
    {
        X = x;
        Y = y;

        CalculateLength();
    }


    // вычисляет вектор по двум точкам
    public static Vector FromPoints(Point start, Point end)
    {
        return new Vector(start.X - end.X, start.Y - end.Y);
    }

    // вычисляет угол между веторами
    public static double CalculateAngleBetween(Vector vector1, Vector vector2)
    {
        if (vector1.Length == 0 || vector2.Length == 0)
        {
            throw new ArgumentOutOfRangeException(vector1.Length == 0 ? nameof(vector1) : nameof(vector2), "Vector can not be with zero lenght");
        }
        var scalarMultiplier = vector1 * vector2;
        var cos = scalarMultiplier / (vector1.Length * vector2.Length);
        var angle = Math.Acos(cos);

        return angle;
    }

    // вычисляет скалярное произведение векторов
    public static double operator *(Vector a, Vector b)
    {
        return a.X * b.X + a.Y * b.Y;
    }


    // вычисляет длину вектора
    private void CalculateLength()
    {
        var xPow2 = X * X;
        var yPow2 = Y * Y;
        var sum = xPow2 + yPow2;
        var sqrt = Math.Sqrt(sum);

        Length = sqrt;
    }
}

在线计算器在这里。

测试！