MA242.003 Day 9 – January 17, 2013 Review: Equations of lines, Section 9.5 Section 9.5 –Planes.

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.

1. Give three non-co-linear points.

Different ways to specify a plane:

Equations of PLANES in space.

1. Give three non-co-linear points.

Different ways to specify a plane:

Equations of PLANES in space.

2. Give two non-parallel intersecting lines.

Different ways to specify a plane:

Equations of PLANES in space.

2. Give two non-parallel intersecting lines.

Different ways to specify a plane:

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

when

their direction vectors are parallel

Two lines are perpendicular

Two lines are perpendicular

when

Two lines are perpendicular

when

their direction vectors are orthogonal

Two planes are parallel

Two planes are parallel

when

Two planes are parallel

when

Their normal vectors are parallel

Two planes are perpendicular

Two planes are perpendicular

when

Two planes are perpendicular

when

Their normal vectors are orthogonal

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