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Distance Between a Point and a Plane

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Planes in Space
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Vocabulary

Complete the chart.
Word Definition
_____________ the 3 dimensional equivalent of a line on a standard rectangular graph
Intercept Form _________________________________________________________
_____________ a vector perpendicular to all possible vectors within a plane
_____________ the angle between two planes in a 3D space
Origin _________________________________________________________
Perpendicular line _________________________________________________________

Planes in Space

Since the normal to the plane is, by definition, perpendicular to all possible vectors within a plane and since the dot product of two vectors is equal to zero for any two perpendicular vectors, we can define a plane in terms of the dot product of the normal vector with any vector, \overrightarrow{v} , within the plane:

\overrightarrow{n} \times \overrightarrow{v} = 0

Which we can also write as

\left \langle n_x, n_y, n_z \right \rangle \times \left \langle (x - x_0), (y - y_0), (z - z_0) \right \rangle = 0

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What is the intercept form of the equation of a plane? _______________________

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What is the equation which specifies the plane in terms of the normal vector and two points on the plane? _______________________

What are the equations of the intercepts of that plane?

 a = ____________    b = ____________     c =____________

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Given the following intersections, write the equation of the plane.

  1. (13, 0, 0), (0, 21, 0) and (0, 0, 17)
  2. (5, 0, 0), (0, 1, 0) and (0, 0, 2)
  3. (27, 0, 0), (0, 12, 0) and (0, 0, 18)
Find the intercepts of the plane given the following equations:
  1. 1x - 7y - z + 10 = 0
  2. -2x + 9y + 4z - 1 = 0
  3. 6x - 11y + 2z + 3 = 0

Use the given equations to determine the normal unit-vector to that plane:

  1. -8x + 7y + 2z + 5 = 0
  2. 10x + 3y - z - 2 = 0
  3. -1x - 2y + 7z + 16 = 0
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Click here for answers.

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Distance Between a Point and a Plane

The position vector for the point closest to a plane is _________________ to the normal vector.

Determine the location of the point on the plane closest to the origin by finding the projection of the given point’s ___________________ onto the _____________________.

The angle between two planes is the same as the angle between their ____________________. 

Use the ___________________ to find this angle.

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The three points define a plane. Determine the point on the plane which is closest to the origin.

  1. P = (3, 8, 10), Q = (-2, 5, 8) and R = (7, 4, 8)
  2. P = (9, -1, 4), Q = (6, 2, -8) and R = (12 , 9, 10)
  3. P = (5, 8,-9), Q = ( -5, 3, 9) and R = (10, 4, -6)


Determine the dihedral angle between each of these planes and the x-y plane, use the |\overrightarrow{n}| you calculated for each plane and recall that the normal to the x-y plane is the unit vector\hat{z} = \left \langle 0, 0, 1 \right \rangle

  1. P = (3, 8, 10), Q = (-2, 5, 8) and R = (7, 4, 8)
  2. P = (9, -1, 4), Q = (6, 2, -8) and R = (12 , 9, 10)
  3. P = (5, 8,-9), Q = ( -5, 3, 9) and R = (10, 4, -6)

Determine the dihedral angle between the two planes.

  1. -7x + 20y + 6z + 4 = 0 and -19x - 3y + z + 5 = 0
  2. 5x - 8y + 20z - 5 = 0 and 6x + y + 19z - 7 = 0
  3. 14x + 11y - 5z - 16 = 0 and 11x - 13y + 8z + 4 = 0

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Click here for answers.

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