MA242.003
•Day 9 – January 17, 2013
•Review: Equations of lines, Section 9.5
•Section 9.5 –Planes
Equations of PLANES in space.
Equations of PLANES in space.
Different ways to specify a plane:
Equations of PLANES in space.
Different ways to specify a plane:
1. Give three non-co-linear points.
Equations of PLANES in space.
Different ways to specify a plane:
1. Give three non-co-linear points.
Equations of PLANES in space.
Different ways to specify a plane:
2. Give two non-parallel intersecting lines.
Equations of PLANES in space.
Different ways to specify a plane:
2. Give two non-parallel intersecting lines.
Equations of PLANES in space.
Different ways to specify a plane:
3. Specify a point
and a normal vector
Given:
Equations of PLANES in space.
Equations of PLANES in space.
Example: Find an equation for the plane containing
the point (1,-5,2) with normal vector <-3,7,5>
Example: Find an equation for the plane containing
the the points P=(1,-5,2), Q=(-3,8,2) and R=(0,-1,4)
REMARK: How equations of planes occur in problems
REMARK: How equations of planes occur in problems
The Geometry of Lines and Planes
• For us, a LINE in space is a
The Geometry of Lines and Planes
• For us, a LINE in space is a
Point
and a direction vector v = <a,b,c>
The Geometry of Lines and Planes
• For us, a Plane in space is a
The Geometry of Lines and Planes
• For us, a Plane in space is a
Point on the plane
And a normal vector n = <a,b,c>
Two lines are parallel
Two lines are parallel
when
Two lines are parallel
their direction
vectors are parallel
when
Two lines are perpendicular
Two lines are perpendicular
when
Two lines are perpendicular
their direction vectors
are orthogonal
when
Two planes are parallel
Two planes are parallel
when
Two planes are parallel
Their normal vectors
are parallel
when
Two planes are perpendicular
Two planes are perpendicular
when
Two planes are perpendicular
Their normal vectors
are orthogonal
when
A line is parallel to a plane
A line is parallel to a plane
when
A line is parallel to a plane
when
the direction vector v for the line is orthogonal to the
normal vector n for the plane
A line is perpendicular
to a plane
A line is perpendicular
to a plane
when
A line is perpendicular
to a plane
when
the direction vector v for the line is parallel to the
normal vector n for the plane
Example Problems
Example Problems
Example Problems
Example Problems