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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, \begin{align*}\overrightarrow{v}\end{align*} , within the plane:

\begin{align*}\overrightarrow{n} \times \overrightarrow{v} = 0\end{align*}

Which we can also write as

\begin{align*}\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\end{align*}

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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?

 \begin{align*}a =\end{align*} ____________    \begin{align*}b =\end{align*} ____________     \begin{align*}c =\end{align*}____________

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

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

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

  1. \begin{align*}-8x + 7y + 2z + 5 = 0\end{align*}
  2. \begin{align*}10x + 3y - z - 2 = 0\end{align*}
  3. \begin{align*}-1x - 2y + 7z + 16 = 0\end{align*}
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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. \begin{align*}P = (3, 8, 10), Q = (-2, 5, 8)\end{align*} and \begin{align*}R = (7, 4, 8)\end{align*}
  2. \begin{align*}P = (9, -1, 4), Q = (6, 2, -8)\end{align*} and \begin{align*}R = (12 , 9, 10)\end{align*}
  3. \begin{align*}P = (5, 8,-9), Q = ( -5, 3, 9)\end{align*} and \begin{align*}R = (10, 4, -6)\end{align*}


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

  1. \begin{align*}P = (3, 8, 10), Q = (-2, 5, 8)\end{align*} and \begin{align*}R = (7, 4, 8)\end{align*}
  2. \begin{align*}P = (9, -1, 4), Q = (6, 2, -8)\end{align*} and \begin{align*}R = (12 , 9, 10)\end{align*}
  3. \begin{align*}P = (5, 8,-9), Q = ( -5, 3, 9)\end{align*} and \begin{align*}R = (10, 4, -6)\end{align*}

Determine the dihedral angle between the two planes.

  1. \begin{align*}-7x + 20y + 6z + 4 = 0\end{align*} and \begin{align*}-19x - 3y + z + 5 = 0 \end{align*}
  2. \begin{align*}5x - 8y + 20z - 5 = 0\end{align*} and \begin{align*}6x + y + 19z - 7 = 0 \end{align*}
  3. \begin{align*}14x + 11y - 5z - 16 = 0\end{align*} and \begin{align*}11x - 13y + 8z + 4 = 0 \end{align*}

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