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MCV4U1-UNIT EIGHT-LESSON FIVE Lesson Five: The Cartesian (or Scalar) Equation of a Plane We can find the scalar equations of a plane using dot product, just as we did for a line: n A, B, C Px, y, z P1x1, y1, z1 Plane Let n A, B, C be normal to the plane. Let Px, y, z be any point on the plane Let P1x1, y1, z1 be a specific point on the plane. n P1P 0 n A, B, C x x1 , y y1 , z z1 Ax Ax1 By By1 Cz Cz 1 0 Let D Ax1 By1 Cz1 Then the scalar equation of a plane is ..... Ax By Cz D 0 Example 1: Find the scalar equation of the plane through the point (1,-1,0) and having normal (2,-1,5). Solution: n P1P 0 2,1,5 x 1, y 1, z 0 2 x 2 y 1 5z 0 2 x y 5z 3 0 Example 2: Find the scalar equation of the plane through the point (3,-1,2) and with direction vectors (2,1,0) and (3,5,2). Solution: n 2,1,0 3,5,2 2,4,13 2,4,13 x 3, y 1, z 2 0 2 x 6 4 y 4 13z 26 0 2 x 4 y 13z 16 0