### NEW SMARANDACHE ALGEBRAIC STRUCTURES

G Waghmare, S More
unpublished
Generally In R' any plane with equation x + y + z = a, where a is nonzero number. is not a linear space under the usual vector addition and scalar multiplication If we define new algebraic operations on the plane x + y + z = a it will become a linear space in R'. The aoditive iOentity of this linear space has nonzero components. The plane x + y + Z = a touches the x-axis at point A (a .0 ,0) • y-axis at point B (a,a,O) and Z-axIS at point C (O,G,a). Take triangle ABC as a fixed equilateral
more » ... ed equilateral triangle Known as " triangle of reference. " From any point P in its plane draw perpendiculars PM, PN and PL to AC, AB and BC respectively. Let).."(P M) = p" 1(p N) = Pz and J. (PL) = PJ. These P, , Pz and P3 are called the trilinear coordinater of point P [ Loney 1, Smith 2, Sen 3 ], The coordInate P, is positive if P and the vertex B of the triangle are on the same side of L AC and P, is negative if P and B are on the opposite sides of AC, So for the other coordinates Pz and P3' 2. Length of each side of the triangle is J2. lai = b (say).