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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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#### 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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