Name _________________
Worksheet 8.8
Lines and Planes Review
1. Parametric Vector Equation for a Line
Previously we found a robust way to define a line using vectors and a parameter (usually t for
time). Let us review this idea in three dimensions.
2
(a) Find a unit vector in the direction of PQ   2 . Make a sketch.
6 
(b) Find the parametric vector equation of the line containing the points
 x  3 
P  (3, 1,  4) and Q  (5, 3, 2) . Ensure that when t  0 , X   y    1   P and that the
   
 z   4
distance from t  0 to any point is always equal to its value of t. This is accomplished by having
a spanning vector (the vector that is multiplied by t) with magnitude of one. Roughly sketch
what you have done.
(c) Notice that if we do not care for the luxury of that last requirement, we do not need the
spanning vector to be a unit vector. Finding an equation becomes much simpler since we do not
need to normalize PQ (make its magnitude equal to one). Find the equation for the line
containing P  (3, 1,  4) and Q  (5, 3, 2) such that when t  0 , X  P , and when t  1, X  Q .
Again, make a quick sketch.
2. Ordinary Equation for a Line
Now let us use vectors in different way to define a line in two dimensions. We will see that this
approach (the one used by your textbook) generalizes to higher dimensions quite differently than
the parametric vector equation. While the parametric method gives us a reliable tool to form a
one dimensional line in space (whether this space was 2-D, 3-D, or some higher dimension), this
method will give us a reliable way of lowering the dimension of our space by one. In other
words, in two dimensions we can form a line, in three dimensions we can form a plane, and so
on.
(a) Sketch the point P   3, 4 , then draw a line such that P lies on the line, and such that P is
closer to the origin that any other point on the line.
(b) Find a vector N which is perpendicular to this line (it does not need to be a unit vector). This
should require minimal calculation. Sketch this vector above.
(c) Mark any point on the line X  ( x, y ) , and draw the vector X  P
(d) No matter what your choice of X, what is always true about ( X  P)  N when X lies on the
given line?
(e) Find the equation for the line you have drawn in ax  by  c form. If you are thoughtful, you
do not have to do anything more than find two dot products in order to accomplish this task.
3. Defining a Plane
What information do we need to define a plane in three dimensions? Let us find out how similar
it is to a line in two dimensions. Include a sketch to answer each of the following.
(a) How do you define a plane using three points? What needs to be true about these points?
(b) How would you define a plane using only two lines? What needs to be true about these
lines?
(c) We can define a plane using given a single point and a single line such that there are no
special cases that fail. Describe how you can do this using a sketch.
4. Equation for a Plane in Three Dimensions
Consider P   x1 , y1 , z1  as a specific point that lies on a given plane. Let X  ( x, y, z ) be any
a 
general point that lies on this plane. Let N  b  be a vector perpendicular to the plane. Also
 c 
 x  x1 
note that the vector that points from P to X is ( X  P)   y  y1 
 z  z1 
(a) Make a three dimensional sketch of this plane, the point P , and an arbitrary point X that is
not equal to P . Also, draw the vectors N and ( X  P) such that the base of both vectors are
coincident (at the same point) with P
(b) Based on your sketch, what is always true about the dot product ( X  P)  N
(c) Use your answer to describe the general form of the equation of a plane in three dimensions
using vectors?
(d) Does this differ in any way from the equation for a line in two dimensions?
(e) Here is a tough one. What is the general equation for the distance from the plane to the
origin? (Hint, make a sketch and think about projections)
5. Some Problems
As review, the key formula that defines a plane in three dimensions (or a line in two dimensions)
is X  N  P  N . You can use this idea to solve these otherwise difficult problems. Recall that
x
X   y 
 z 
3
4


(a) Find an equation of a plane through P   1 which is perpendicular to the N   1
 5 
 2 
 x  1   3 
(b) Find an equation of the plane which is perpendicular to the line  y    2  t  2 at the
 z   3   6 
point when t = 2
(c) Find the equation of the plane which is equidistant from the points
R  (5, 3, 2) and Q  (3, 1,  4)
Bonus. Parametric Plane
Can we use a single parametric vector equation to define a plane in three dimensions? If we
allow ourselves to use more than one parameter, then the answer is yes.
(a) Given the points H  (2,  5, 1), K  (4,  6, 1), and L  (3,  4, 2) , find a vector perpendicular
to both HK and HL (There is no quick way to do this, but if you play around for a while, you
can find a vector N such that HK  N  0 and HL  N  0 )
(b) Find the equation (in the form ax  by  cz  d ) for the plane that contains all three of these
points.
(c) Given two points H and K, we have learned how to define a line with the equation
X  H  tHK . Give a parametric vector equation for the plane that contains H, K, and L. You
will need two parameters t1 and t2 , not just one.
1
(d) Use part (b) to verify that the point  3 lies on this plane.
 2 
1
(e) Given your answer in part (c), find what values of t1 and t2 put you at  3
 2 