実際にはPlaneとRayだけでやれる。
軸が違うので変換or変数入れ替えしないといけない。
(このコードでは高さがz奥行きがy、幅がx,unityでは高さy,奥行きz,幅x向きも違う。)


using UnityEngine;

public class StraightLine
{
    public readonly float XParamCoefficient;
    public readonly float XConstant;
    public readonly float YParamCoefficient;
    public readonly float YConstant;
    public readonly float ZParamCoefficient;
    public readonly float ZConstant;

    public StraightLine(float XParamCoefficient, float XConstant,
                        float YParamCoefficient, float YConstant,
                        float ZParamCoefficient, float ZConstant)
    {
        this.XParamCoefficient = XParamCoefficient;
        this.XConstant = XConstant;
        this.YParamCoefficient = YParamCoefficient;
        this.YConstant = YConstant;
        this.ZParamCoefficient = ZParamCoefficient;
        this.ZConstant = ZConstant;
    }

    public static StraightLine Make(Vector3 p1, Vector3 p2)
    {
        var d = p2 - p1;
        return new StraightLine(d.x, p1.x, d.y, p1.y, d.z, p1.z);
    }
}

using UnityEngine;

public sealed class Plane
{
    private readonly float p;
    private readonly float q;
    private readonly float r;

    private readonly float x0;
    private readonly float y0;
    private readonly float z0;

    private Plane(float p, float q, float r, float x0, float y0, float z0)
    {
        this.p = p;
        this.q = q;
        this.r = r;
        this.x0 = x0;
        this.y0 = y0;
        this.z0 = z0;
    }

    public static Plane Make(Vector3 a, Vector3 b, Vector3 c)
    {
        var ab = b - a;
        var ac = c - a;
        var n = new Vector3(ab.y * ac.z - ab.z * ac.y,
                            ab.z * ac.x - ab.x * ac.z,
                            ab.x * ac.y - ab.y * ac.x);

        return new Plane(n.x, n.y, n.z, a.x, a.y, a.z);
    }

    public Vector3 GetIntersection(float XParamCoefficient, float XConstant,
                                   float YParamCoefficient, float YConstant,
                                   float ZParamCoefficient, float ZConstant)
    {
        var param = (p * (x0 + XConstant) + q * (y0 + YConstant) + r * (z0 + ZConstant))
                    / (p * XParamCoefficient + q * YParamCoefficient + r * ZParamCoefficient);

        return new Vector3(XParamCoefficient * param - XConstant,
                           YParamCoefficient * param - YConstant,
                           ZParamCoefficient * param - ZConstant);
    }
}