Planes Notes (9/26) (copy)

Plane Notes (9/26)

Linear Equation of a Plane in R³
- General form: ( ax + by + cz = d )
- A plane in space is defined as the set of all terminal points of vectors emanating from a given point ( P_0 ).
- Normal vector ( n = (a, b, c) ) to the plane.
- Point on the plane: ( P = (x, y, z) )
- Scalar equation derived from point ( (x_0, y_0, z_0) ):
  - ( a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 )
Activity 1.5.4
- (a) Write a scalar equation of plane ( P_1 ) through point ( (0, 2, 4) ) and perpendicular to vector ( n = (2, -1, 1) ).
- (b) Determine if point ( (2, 0, 2) ) is on plane ( P_1 ).
- (c) Write a scalar equation of plane ( P_2 ) parallel to ( P_1 ) and passing through point ( (3, 0, 4) ).
  - Hint: Compare normal vectors of the planes.
- (d) Write a parametric description of line ( l ) passing through point ( (2, 0, 2) ) and perpendicular to plane ( P_3 ) given by ( x + 2y - 2z = 7 ).
- (e) Find the intersection point of line ( l ) with plane ( P_3 ).

Activity 1.5.5
- Given points: ( P = (1, 2, -1) ), ( P_1 = (1, 0, -1) ), ( P_2 = (0, 1, 3) )
- (a) Determine vectors ( PP_1 ) and ( PP_2 ):
  - ( PP_1 = (0, -2, 0) )
  - ( PP_2 = (-1, -1, 4) )
- (b) Find a normal vector ( n ) to plane ( p ) using the cross product:
  - Resulting normal vector: ( (-8, 0, -2) )
- (c) Find a scalar equation of plane ( p ).
- (d) Consider a second plane ( q ) with scalar equation:
  - ( -1 - 14 - 3(x - 1) + 4(y + 3) + 2(z - 5) = 0 )
  - Find two points on plane ( q ) and a normal vector ( m ).
- (e) The angle between two planes is the acute angle between their normal vectors.
  - Calculate the angle between planes ( p ) and ( q ):
    - Use the formula ( \cos(\theta) = \frac{n_1 \cdot n_2}{|n_1||n_2|} )
- Intersection of a Plane with a Line
  - The intersection of a plane with a line is a point or a line, referred to as traces.
  - Parallel planes do not intersect.