<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# Distance Between a Point and a Plane

## Points closest to the origin and Dihedral Angles.

Estimated10 minsto complete
%
Progress
Practice Distance Between a Point and a Plane

MEMORY METER
This indicates how strong in your memory this concept is
Progress
Estimated10 minsto complete
%
Planes in Space

Feel free to modify and personalize this study guide by clicking “Customize.”

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

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

Which we can also write as

nx,ny,nz×(xx0),(yy0),(zz0)=0\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*}

.

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?

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

.

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

1. (13,0,0),(0,21,0)\begin{align*}(13, 0, 0), (0, 21, 0)\end{align*} and (0,0,17)\begin{align*}(0, 0, 17)\end{align*}
2. (5,0,0),(0,1,0)\begin{align*}(5, 0, 0), (0, 1, 0)\end{align*} and (0,0,2)\begin{align*}(0, 0, 2)\end{align*}
3. (27,0,0),(0,12,0)\begin{align*}(27, 0, 0), (0, 12, 0)\end{align*} and (0,0,18)\begin{align*}(0, 0, 18)\end{align*}
Find the intercepts of the plane given the following equations:
1. 1x7yz+10=0\begin{align*}1x - 7y - z + 10 = 0\end{align*}
2. 2x+9y+4z1=0\begin{align*}-2x + 9y + 4z - 1 = 0\end{align*}
3. 6x11y+2z+3=0\begin{align*}6x - 11y + 2z + 3 = 0\end{align*}

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

1. 8x+7y+2z+5=0\begin{align*}-8x + 7y + 2z + 5 = 0\end{align*}
2. 10x+3yz2=0\begin{align*}10x + 3y - z - 2 = 0\end{align*}
3. 1x2y+7z+16=0\begin{align*}-1x - 2y + 7z + 16 = 0\end{align*}
.

.

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

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

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

Determine the dihedral angle between the two planes.

1. 7x+20y+6z+4=0\begin{align*}-7x + 20y + 6z + 4 = 0\end{align*} and 19x3y+z+5=0\begin{align*}-19x - 3y + z + 5 = 0 \end{align*}
2. 5x8y+20z5=0\begin{align*}5x - 8y + 20z - 5 = 0\end{align*} and 6x+y+19z7=0\begin{align*}6x + y + 19z - 7 = 0 \end{align*}
3. 14x+11y5z16=0\begin{align*}14x + 11y - 5z - 16 = 0\end{align*} and 11x13y+8z+4=0\begin{align*}11x - 13y + 8z + 4 = 0 \end{align*}

.