1. No category

UNIT 8: EQUATIONS OF LINES AND PLANES
8.1 VECTOR AND PARAMETRIC EQUATIONS OF A LINE IN R2
To find vector and parametric equations of lines, we must be given the following:
1) Two distinct points, or
2) One point and a vector that defines the direction of the line
Def. A direction vector is a nonzero vector m
= a, b parallel/collinear to the given line. This vector is
represented with its tail at the origin and its head at (a,b). The x and y components of this direction
vector are called its direction numbers.
Because there are an infinite number of vectors that are collinear to a given line, we use a parameter, t,
in our vector equation.
The vector equation of a line in R2 is: r = r + tm
, t ЄR. It can also be written as (x, y) = (x , y +
ta, b.
r is the vector from the origin to any point on the line, and m
is the direction vector of the line.
The parametric form of the equation breaks down the line into its components:
x = x + taand y = y + tb, tЄR.
Ex. A) Determine the vector and parametric equations of the line passing through point A(1, 4) with
direction vector m
= −3, 3.
B) sketch the line, and determine the coordinates of 4 points on the line.
C) Is either point Q(-21, 23) or R(-29, 34) on the line?
Ex. A)Determine vector and parametric equations for the line containing points E(-1,5) and F(6,11).
B) what are the coordinates of this point where the line crosses the x-axis?
C) Can the equation r = −15, −7+ t(14/3, 4), tЄR, also represent the line containing point E and F?
Ex. Determine a vector equation for the line perpendicular to r = 4, 1 + s−3, 2, sЄR,and passes
through the point P(6,5).
8.2 CARTESIAN EQUATION OF A LINE
Recall that the slope of a line is defined as
!"
=
#$ %#&
.
'$ %'&
This can be connected to what we have learned
in chapter 1, as a direction vector can also tell us about the “slope”, or direction of the line. Given two
= m
points on the line, A and B, AB
= x* − x+ , y* − y+ . This direction vector is equivalent to a vector
with its tail at the origin and its head at x* − x+ , y* − y+ .
,
If the direction vector of a line is m
= a, b, then the slope of the line is m = -.
Ex. Determine the equivalent vector and parametric equations of the line y = ¾ x + 2.
Ex. For the line with equation r = 3, −6 + t−1, −4, tЄR, determine the equivalent slope/y-int
form.
Ex. Determine the Cartesian form of the line with the equation r = 1,4 + s0,2, sЄR.
Def. The normal to a line L is the perpendicular line to L, drawn from the origin. It is perpendicular to
any vector on L. We denote the normal to any line as n = A, B.
In R2, the Cartesian equation of a line is given by Ax + By + C = 0, where (A,B) is the normal to this
line. A normal to this line is a vector drawn from the origin perpendicular to the given line to the
point N(A,B).
Ex. Determine the Cartesian equation of the line passing through A(4, -2) which has n = 5, 3 as its
normal.
We can use our knowledge of normals now to extend into parallel and perpendicular lines.
If lines L+ andL* have normals andn
n+
* respectively,
The two lines are parallel if and only if (iff) their normals are scalar multiples (n
+ =
kn
,
* kЄR, k ≠ 0. The lines’ direction vectors are also scalar multiples.
2. The two lines are perpendicular iff the dot product of their normals is 0. The dot product
of the direction vectors will also be 0.
1.
Ex.
a) Show that the lines L+ : 3x − 4y − 6 = 0 and L* : 6x − 8y + 12 = 0 are parallel and noncoincident.
b) For what value of k are the lines L3: kx + 4y – 4 = 0 and L4: 3x – 2y – 3 = 0 perpendicular?
Ex. Determine the acute angle formed at the point of intersection created by the following lines:
L+ : x, y = 2,2 + s−1,3, sЄR
L* : x, y = 5,1 + t3,4, tЄR
8.3 VECTOR, PARAMETRIC AND SYMMETRIC EQUATIONS OF LINES IN R3
We can express lines in R3 similarly in vector and parametric form.
Vector Equation: r = r + tm
, tЄR where risthe
vector from the origin to any point on the line and m
is a direction vector.
Parametric Equation: x = x +ta, y = y + tb, z = z + tc
NB: m
= a, b, c
In R3 there is another way to express the equation of a line. This is called the symmetric equation for a
line in R3. To find the symmetric equation, we take the parametric equations above and isolate for t in
each.
:=
; − ; = − = ? − ?
=
=
<
>
@
The same variables apply as above: (x , y , z is the vector from the origin to a point on the line, and
(a,b, c) is a direction vector of the line.
Ex.
a) Write the symmetric equations of the line passing through the points A(-1, 5, 7) and B(3, -4, 8).
b) Write the symmetric equations of the line passing through the points P(-2, 3, 1) and Q(4, 3, -5).
c) Show that the following are vector equations for different lines:
L+ = 1, 6, 1 + s−1, 1, 2
1 1
L* = −3, 10, 12 + t , − , −1
2 2
8.4 VECTOR AND PARAMETRIC EQUATIONS OF A PLANE
Recall: planes are flat surfaces that extend infinitely far in all directions. We will represent planes with
the symbol π.
Planes can be determined in four different ways:
a)
b)
c)
d)
A line and a point not on the line
Two intersecting lines
Three non-collinear points
Two parallel and non-coincident lines
●
●
●
●
Given a set of these data,we can construct a unique plane that contains them.
In R3 the vector equation of a plane is
, where r = (x , y , z and x , y , z is a point on the plane, and
r = r + sa + tb
are two noncollinear vectors.
aandb
Accordingly, the parametric equations of a plane are
x = x + sa+ + tb+ , y = y +sa* + tb* , z = z + saA + tbA . For all of these, s and t ЄR.
Ex.
a) Determine a vector equation and parametric equations for the plane that contains the points
A(-1,3, 8), B(-1, 1, 0) and C(4, 1, 1).
b) Do either of the points P(14, 1, 3) or Q(14, 1, 5) lie on this plane?
Ex. A plane π has r = 6, −2, −3 + s1, 3,0 + t2, 2, −1, s, tЄR,as its equation. Determine the point
of intersection between π and the z-axis.
Ex. Determine the vector and parametric equations for the plane containing the point P(1, -5, 9) and the
line r = 1, 1, 1 + s−1, 1, 0, sЄR.
8.5 THE CARTESIAN EQUATION OF A PLANE
The Cartesian equation for a plane in R3 is of the form Ax + By + Cz + D = 0, where the normal to the
plane is n = A, B, C.The normal is a non-zero vector that extends from the origin to the plane, and is
perpendicular to all vectors in the plane.
andn
*
Consider perpendicular planes: if π+ andπ* are two perpendicular planes, with normals n
+
respectively, then their normals are perpendicular. Their dot product will be 0.
For parallel planes, their normals will be parallel (n
+ = kn
* ) for all non-zero real numbers k.
To find the angle between two intersecting planes, we shall use what we know about the dot product,
and the normal to each plane.
cos F =
G
●G
+ *
|
HG
+ |HG
*
Ex. The point A(1, 2, 2) is a point on the plane with normal n = −1, 2, 6. Determine the Cartesian
equation of the plane.
Ex. Determine the Cartesian equation of the plane containing the points A(-1, 2, 5), B(3, 2, 4) and
C(-2, -3, 6).
Ex. A) Determine the Cartesian form of the plane whose equation in vector form is r = 1, 2, −1 +
s1, 0, 2 + t−1, 3, 4, s, tЄR.
b) Determine the vector and parametric equations of the plane with Cartesian equation
x – 2y + 5z – 6 = 0.
Ex. A) Show that the planes π+ : 2x − 3y + z − 1 = 0 and π* : 4x − 3y − 17z = 0 are perpendicular.
b) show that the planes πA : 2x − 3y + 2z − 1 = 0 and πJ : 2x − 3y + 2z − 3 = 0 are parallel but not
coincident.
Ex. Determine the angle between the two planes π+ : x − y − 2z + 3 = 0 and π* : 2x + y − z + 2 = 0.