object Vec {
  val X_AXIS = Vec(1,0)
  val Y_AXIS = Vec(0,1)
}

case class Vec(x: Double, y: Double) {
  def dotProduct(v: Vec) = x*v.x+y*v.y
  def crossProduct(v: Vec) = x*v.y-y*v.x;
  def length = math.sqrt(x*x+y*y)
  def rotate(theta: Double) = {
      val sin = math.sin(theta);
      val cos = math.cos(theta);
      Vec(x*cos-y*sin,
          x*sin+y*cos);
  }
  def rotate90 = Vec(-y, x)
  def +(v: Vec) = Vec(x+v.x, y+v.y)
  def -(v: Vec) = Vec(x-v.x, y-v.y)
  def *(d: Double) = Vec(x*d, y*d)
  def /(d: Double) = Vec(x/d, y/d)
  def unary_- = Vec(-x, -y)
  override def toString = f"Vec($x%.3f, $y%.3f)"
}

object Matrix {
  def rotationMatrix(theta: Double) = {
    val v = Vec.X_AXIS.rotate(theta)
    Matrix(v, v.rotate90)
  }
  def apply(a: Double, b: Double,
            c: Double, d: Double): Matrix = Matrix(Vec(a,c), Vec(b,d))
}

case class Matrix(c1: Vec, c2: Vec) {
  def *(v: Vec) = c1*v.x + c2*v.y
  def *(d: Double) = Matrix(c1*d, c2*d)
  def determinant = c1.crossProduct(c2)
  def inverse = Matrix( c2.y, -c2.x,
                       -c1.y,  c1.x) * (1/determinant)
  override def toString = f"Matrix(${c1.x}%.3f, ${c2.x}%.3f\n" +
                          f"       ${c1.y}%.3f, ${c2.y}%.3f)"
}

case class CoordinateSystem(origin: Vec, xAxis: Vec, yAxis: Vec) {
  val matrix = Matrix(xAxis, yAxis)
  val inverseMatrix = matrix.inverse
  def translate(v: Vec) = matrix*v + origin
  def translateBack(v: Vec) = inverseMatrix*(v-origin)
}

val cs = CoordinateSystem(Vec(500,100),
                          Vec(1,1),
                          Vec(-1,1))
val v = Vec(2,3)
val v2 = cs.translate(v)
println(v2)
println(cs.translateBack(v2))