*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, , 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.**

- and
- and
- and

**Find the intercepts of the plane given the following equations:**

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

**.**

#### 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.**

- and
- and
- 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**

- and
- and
- and

**Determine the dihedral angle between the two planes.**

- and
- and
- and

.

Click here for answers.