Chapter 9
Planes, Determinanats, and Vectors

9.1 Planes

• Equation:

  Ax + By + Cz + D = 0

• Parallel planes differ only in D

• Normal to a plane:

  (x,y,z) = (A,B,C)

9.2 Vectors

• Vector: a difference V = (x,y,z) between points.

• Length of a vector |V | =

• Unit vector in the direction of V: V _ |V |

• Addition:

  V ₁ + V ₂ = (x₁ + x₂,y₁ + y₂,z₁ + z₂)

• Subtraction:

  V ₁ - V ₂ = (x₁ - x₂,y₁ - y₂,z₁ - z₂)

• INNER / SCALAR / DOT Product:

  V ₁ ^. V ₂ =x₁ x₂ + y₁ y₂ + z₁z₂ =|V ₁ ||V ₂ |cos

  • |V | =

  • Angle between two vectors cos = V ₁^.V ₂ __ |V ₁||V ₂|

  • Orthotogonal vectors: V ₁ ^. V ₂ = 0

  • Points Q in the plane that contains P and is perpendicular to V :

    (Q - P) ^. V = 0.

  • Normal to surface F(x,y,z) = 0 at point (x,y,z): ^T
• CROSS / VECTOR Product V ₁ × V ₂

  

  • V ₁ × V ₂ perpendicular to the plane of V ₁ and V ₂ forming a right-hand system.

  • |V ₁ × V ₂ | = |V ₁ ||V ₂ | sin

9.3 Determinants

• Determinant of order n.

• Minor determinant M_ij of D: Determinant D without row i and column j

• Determinants have values.
  • Order = 1: |a| = a

  • Order > 1:

    a₁₁M₁₁ - a₁₂M₁₂ + a₁₃M₁₃ - a₁₄M₁₄ +

  • Order 2.

    = a₁₁a₂₂ - a₁₂a₂₁

  • Order 3.

    = a₁₁M₁₁ -a₁₂M₁₂ +a₁₃M₁₃

• Plane equation from 3 noncolinear points

  

9.4 Properties of Determinants

• A determinant keeps its value if row_i interchanges with column_i for all i

  =

• Interchange 2 rows (or columns) multiply determinant by -1

  = -

• Multiply elements of a row (or a column) by a constant multiply determinant by the constant

  = c

• Adding a line multiplied by a constant to another line does not change the value of the det.

=