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Distance Between a Point and a Plane

Points closest to the origin and Dihedral Angles.

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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,  , within the plane:

Which we can also write as

.

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

.

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?

  ____________     ____________     ____________

.

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

  1.  and 
  2.  and 
  3.  and 
Find the intercepts of the plane given the following equations:

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

.
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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.

.

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

  1.  and 
  2.  and 
  3.  and 


Determine the dihedral angle between each of these planes and the x-y plane, use the  you calculated for each plane and recall that the normal to the x-y plane is the unit vector

  1.  and 
  2.  and 
  3.  and 

Determine the dihedral angle between the two planes.

  1.  and 
  2.  and 
  3.  and 

.

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