with normal vector n =
is found by stating thatthe dot product between any vector in the plane and the normal vector = 0.
| LINES AND PLANES IN 3-SPACE |
Notes and examples
Topics Covered in this Lesson
Point/Normal Equation of a Plane
Equation of a Plane Given 3 Points in the Plane
PARAMETRIC Equations of a Line in R3
Symmetric Equations of a Line in R3
Distance from a Point to a Plane
Point of Intersection Between a Line and a Plane
Parametric Equations for the Line of Intersection of 2 Planes
.
.
.
Point/Normal Equation of a Plane
A plane is defined by a vector perpendicular to every
vector in the plane called the NORMAL.
The equation of the plane passing through P(x1, y1, z1)
with normal vector n = is found by stating that
the dot product between any vector in the plane and the normal vector = 0.
ie: 
or ax + by + cz + d = 0 where d = – (ax1 + by1 + cz1)
.
Equation of a Plane Given 3 Points in the Plane
To find the equation of a plane passing through 3 given points,
find the cross product of two vectors defined by the points
then use it as the normal of the plane. Use any one of the 3 points
and the point/normal approach to find the equation of the plane.
Note: remember that the normal is perpendicular to all vectors in the plane,
so the dot product = 0.
Example 1: Find an equation for the plane passing through
P (1, 2, 1), Q (–2, 3, –1) and R (1, 0, 4).
Solution: The vector PQ = 
, vector PR = 
.
Since these 2 vectors are in the plane, their cross product will be orthogonal to the plane.
. Now we dot this vector with 
and set it = 0.– (x – 1) + 9 (y – 2) + 6 (z – 1) = 0 in point/normal form and
– x + 9y + 6z – 23 = 0 in general form.
.
PARAMETRIC Equations of a Line in R3:
The parametric equations of the line through P(x1, y1, z1)
with direction vector v = 
are found by stating that
any vector on the line is parallel to v.(ie: a multiple of v)
ie: 
, where t c R.
Note: In the parametric equations, we see the components of the direction vector
and the coordinates of the point P.
So, the parametric equations of a line display both a vector and a point.
Example 2: Find the parametric equations of the line through P(–1, –1, 3),
parallel to 
.
Solution: A vector between any point on the line and P is 
.
Since this must be // to v, it must be a multiple of v.
So, 
Note: we see the point P and vector v.
.
.
Symmetric Equations of a Line in R3:
The symmetric equations of the line through P(x1, y1, z1)
with direction vector v = 
are found by eliminating
the parameter t in the parametric equations.
ie: 
,
use them to find the equations of 2 planes that intersect in the line.
So the symmetric equations of the line in the example above will be:
and the equations of 2 planes that intersect in this line will be found
by setting 2 pairs of these expressions = to each other to give:
x + 1 = 2y + 2 or x – 2y – 1 = 0 and
2y + 2 = z – 3 or 2y – z + 5 = 0
.
Distance from a Point to a Plane:
To find the perpendicular or shortest distance from a point P(x0, y0, z0) to the plane
ax + by + cz + d = 0 use the formula:
Example 3: Find the perpendicular distance from P (1, –2, 3) to the plane 2x – y – z – 6 = 0.
D = 
.
Point of Intersection Between a Line and a Plane
Simply replace the x, y, and z in the plane's equation with the parametric expressions for
x, y and z from the line, solve for t then evaluate the point from the parametric equations.
Example 4: Find all points of intersection of the plane 3x + 4y + 5z = 60 and the line with parametric equations
| x = 5 + t | y = –2 + 2t | z = 7 – t |
Solution: 3(5 + t) + 4(– 2 + 2t) + 5(7 – t) = 60 t 6t = 18, so t = 3.
The point is:
x = 5 + 3, y = –2 + 2(3), z = 7 – 3 or ( 8, 4, 4 ).
.
Parametric Equations for the Line of Intersection of 2 Planes:
We solve the system of 2 equations in 3 unknowns.
We need a parameter for our solutions so we get parametric equations.
Example 5: Find parametric equations for the line of intersection
of the planes 2x – y – z = 3 and x + 2y + 3z = 7.
Solution: 
so the parametric equations are:
.
To Prove Planes are Parallel: Show that their normals are multiples of each other.
.
To Prove a Line is Parallel to a Plane: Show that the line's direction vector is z the normal.
.
To Prove Planes are Orthogonal: Show their normals are orthogonal. (dot product = 0 )
.
for practice on these topics do questions 11 and 12
in assign 3
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