right angles, so the four points C, D, H, F lie on a circle, and (by the intersecting secants theorem) ||BF|| ||BC|| = ||BH|| ||BD||, that is ab sin(FBG) = ||BH|| ||r|| sin(FBG), 2(area of triangle BFG) = ab sin(FBG) = ||BH|| ||FG|| = ||BH|| ||r|| sin(FBG), ||m|| = ||FG|| = ||r|| sin(FBG) = ||r|| ||d||, check direction and m = r × d. ∎ Proof 2: Let a ⋅ a = b ⋅ b = 1. This implies that a = −||BE||, b = −||BC||. According to the vector triple product formula, r × (a × b) = (r ⋅ b) a − (r ⋅ a) b Then When ||r|| = 0, the line L passes the origin with direction d. If ||r|| > 0, the line has direction d; the plane that includes the origin and the line L has normal vector m; the line is tangent to a circle on that plane (normal to m and perpendicular to the plane of the picture) centered at the origin and with radius ||r||. Example. Let a0 = 2, a = (−1,0,0) and b0 = −7, b = (0,7,−2). Then (d:m) = (0:−2:−7:−7:14:−4). Although the usual algebraic definition tends to obscure the relationship, (d:m) are the Plücker coordinates of L. Algebraic definition Primal coordinates In a 3-dimensional projective space , let be a line through distinct points and with homogenous coordinates and . The Plücker coordinates are defined as follows: {| |- | || |} (the skew symmetric matrix whose elements are pij is also called the Plücker matrix ) This implies pii = 0 and pij = −pji, reducing the possibilities to only six (4 choose 2) independent quantities. The sextuple is uniquely determined by L up to a common nonzero scale factor. Furthermore, not all six components can be zero. Thus the Plücker coordinates of L may be considered as homogeneous coordinates of a point in a 5-dimensional projective space, as suggested by the colon notation. To see these facts, let M be the 4×2 matrix with the point coordinates as columns. The Plücker coordinate pij is the determinant of rows i and j of M. Because x and y are distinct points, the columns of M are linearly independent; M has rank 2. Let M′ be a second matrix, with columns x′ and y′ a different pair of distinct points on L. Then the columns of M′