3. Parametric Equations of Lines
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
4. Parametric Equations of Lines
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Scalar Multiplication
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ.
5. Parametric Equations of Lines
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Scalar Multiplication
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
6. Parametric Equations of Lines
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Scalar Multiplication –2v –v
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
7. Parametric Equations of Lines
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Scalar Multiplication –2v –v
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
The Base to Tip Rule
Given two vectors u and v, u + v is the
new vector formed by placing the base
of one vector at the tip of the other.
8. Parametric Equations of Lines
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Scalar Multiplication –2v –v
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
The Base to Tip Rule
Given two vectors u and v, u + v is the u
new vector formed by placing the base v
of one vector at the tip of the other.
The Base to Tip Rule
9. Parametric Equations of Lines
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Scalar Multiplication –2v –v
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
The Base to Tip Rule
Given two vectors u and v, u + v is the u
v
new vector formed by placing the base v
of one vector at the tip of the other.
The Base to Tip Rule
10. Parametric Equations of Lines
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Scalar Multiplication –2v –v
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
The Base to Tip Rule
Given two vectors u and v, u + v is the u
v
new vector formed by placing the base v u+v
of one vector at the tip of the other.
The Base to Tip Rule
11. Parametric Equations of Lines
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Scalar Multiplication –2v –v
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
The Base to Tip Rule
Given two vectors u and v, u + v is the u
v
new vector formed by placing the base v u+v
of one vector at the tip of the other.
Again, we begin with the 2D version The Base to Tip Rule
then generalize the results to 3D.
12. Parametric Equations of Lines
Let D = <a, b> be a vector that indicates direction as
shown.
D=<a,b>
x
13. Parametric Equations of Lines
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D.
tD=<ta,tb>
D=<a,b>
x
14. Parametric Equations of Lines
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers,
we get all the points in the "direction" of D,
i.e. the entire line that coincides with D.
tD=<ta,tb>
D=<a,b>
x
15. Parametric Equations of Lines
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers,
we get all the points in the "direction" of D,
i.e. the entire line that coincides with D.
Hence x(t) = at, y(t) = bt tD=<ta,tb>
is a set of parametric
equations for the line through
the origin in the direction of D =
<a, b>. D=<a,b>
x
16. Parametric Equations of Lines
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers,
we get all the points in the "direction" of D,
i.e. the entire line that coincides with D.
Hence x(t) = at, y(t) = bt tD=<ta,tb>
is a set of parametric
equations for the line through P=<c, d>
the origin in the direction of D =
<a, b>. P = (c, d) specifies a
Suppose D=<a,b>
base point,
x
17. Parametric Equations of Lines
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers,
we get all the points in the "direction" of D,
i.e. the entire line that coincides with D.
Hence x(t) = at, y(t) = bt tD=<ta,tb>
is a set of parametric
equations for the line through P=<c, d>
the origin in the direction of D =
<a, b>. P = (c, d) specifies a
Suppose D=<a,b>
base point, using the base-to-tip
addition rule, the vector sums x
P + tD = <c, d> + t<a, b>
18. Parametric Equations of Lines
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers,
we get all the points in the "direction" of D,
i.e. the entire line that coincides with D.
Hence x(t) = at, y(t) = bt tD=<ta,tb>
is a set of parametric
equations for the line through P=<c, d>
the origin in the direction of D =
<a, b>. P = (c, d) specifies a
Suppose D=<a,b>
base point, using the base-to-tip
addition rule, the vector sums x
P + tD = <c, d> + t<a, b>
19. Parametric Equations of Lines
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers, <at+c,bt+d>
we get all the points in the "direction" of D,
i.e. the entire line that coincides with D.
Hence x(t) = at, y(t) = bt tD=<ta,tb>
is a set of parametric
equations for the line through P=<c, d>
the origin in the direction of D =
<a, b>. P = (c, d) specifies a
Suppose D=<a,b>
base point, using the base-to-tip
addition rule, the vector sums x
P + tD = <c, d> + t<a, b>
20. Parametric Equations of Lines
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers, <at+c,bt+d>
we get all the points in the "direction" of D,
L
i.e. the entire line that coincides with D.
Hence x(t) = at, y(t) = bt tD=<ta,tb>
is a set of parametric
equations for the line through P=<c, d>
the origin in the direction of D =
<a, b>. P = (c, d) specifies a
Suppose D=<a,b>
base point, using the base-to-tip
addition rule, the vector sums x
P + tD = <c, d> + t<a, b>= <c + ta, d + tb>
viewed as points, are all the points on the line L.
21. Parametric Equations of Lines
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b>
P=<c, d> D=<a,b>
x
22. Parametric Equations of Lines
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b>
L
P=<c, d> D=<a,b>
x
23. Parametric Equations of Lines
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number.
L
P=<c, d> D=<a,b>
x
24. Parametric Equations of Lines
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number.
L
tD=<ta,tb>
P=<c, d> D=<a,b>
x
25. Parametric Equations of Lines
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number. <at+c,bt+d>
L
tD=<ta,tb>
P=<c, d> D=<a,b>
x
26. Parametric Equations of Lines
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number. <at+c,bt+d>
L
tD=<ta,tb>
P=<c, d> D=<a,b>
x
27. Parametric Equations of Lines
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number. <at+c,bt+d>
This is a vector–equation of L.
L
tD=<ta,tb>
P=<c, d> D=<a,b>
x
28. Parametric Equations of Lines
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number. <at+c,bt+d>
This is a vector–equation of L.
If (x, y) is a generic point on L then L
(x, y) = (c, d) + t*(a, b) = tD=<ta,tb>
(c + at, d + bt).
P=<c, d> D=<a,b>
x
29. Parametric Equations of Lines
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number. <at+c,bt+d>
This is a vector–equation of L.
If (x, y) is a generic point on L then L
(x, y) = (c, d) + t*(a, b) = tD=<ta,tb>
(c + at, x = c + at, y = d + bt is a set of
Hence d + bt).
parametric equations of L. P=<c, d> D=<a,b>
x
30. Parametric Equations of Lines
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number. <at+c,bt+d>
This is a vector–equation of L.
If (x, y) is a generic point on L then L
(x, y) = (c, d) + t*(a, b) = tD=<ta,tb>
(c + at, x = c + at, y = d + bt is a set of
Hence d + bt).
parametric equations of L. P=<c, d> D=<a,b>
Example A. Find a set of
x
parametric equations for the line L
a. in the direction of D = <2, 1>,
passing through P = (4, 5).
31. Parametric Equations of Lines
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number. <at+c,bt+d>
This is a vector–equation of L.
If (x, y) is a generic point on L then L
(x, y) = (c, d) + t*(a, b) = tD=<ta,tb>
(c + at, x = c + at, y = d + bt is a set of
Hence d + bt).
parametric equations of L. P=<c, d> D=<a,b>
Example A. Find a set of
x
parametric equations for the line L
a. in the direction of D = <2, 1>,
passing through P = (4, 5). P=(4, 5)
D=<2, 1>
x
32. Parametric Equations of Lines
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number. <at+c,bt+d>
This is a vector–equation of L.
If (x, y) is a generic point on L then L
(x, y) = (c, d) + t*(a, b) = tD=<ta,tb>
(c + at, x = c + at, y = d + bt is a set of
Hence d + bt).
parametric equations of L. P=<c, d> D=<a,b>
Example A. Find a set of
x
parametric equations for the line L
( 2t + 4, 1t + 5)
a. in the direction of D = <2, 1>,
passing through P = (4, 5). P=(4, 5)
The parametric equations of L are D=<2, 1>
x(t) = 2t + 4, y(t) = 1t + 5. x
33. Parametric Equations of Lines
Line Equations R3
Let D = <a, b, c> be a vector as shown,
the set of points tD =(at, bt, ct)
form the line through the origin
coinciding with the vector D or
that it’s in the same direction as
D.
34. Parametric Equations of Lines
Line Equations R3
Let D = <a, b, c> be a vector as shown,
the set of points tD =(at, bt, ct) z
form the line through the origin
coinciding with the vector D or that
it’s in the same direction as D. tD=<at,bt,ct>
D = <a, b, c> y
x
35. Parametric Equations of Lines
Line Equations R3
Let D = <a, b, c> be a vector as shown.
the set of points tD =(at, bt, ct) z
form the line through the origin
P=(d,e,f)
coinciding with the vector D or that
it’s in the same direction as D. tD=<at,bt,ct>
Let P = (d, e, f) be a D = <a, b, c> y
point in space,
x
36. Parametric Equations of Lines
Line Equations R3
Let D = <a, b, c> be a vector as shown.
the set of points tD =(at, bt, ct) z
P+tD
form the line through the origin
P=(d,e,f)
coinciding with the vector D or that
it’s in the same direction as D. tD=<at,bt,ct>
Let P = (d, e, f) be a D = <a, b, c> y
point thespace,
then in line containing P,
in the direction of D consists x
of points P + tD = (d + at, e + bt, f + ct ).
37. Parametric Equations of Lines
Line Equations R3
Let D = <a, b, c> be a vector as shown.
the set of points tD =(at, bt, ct) z
P+tD
form the line through the origin
P=(d,e,f)
coinciding with the vector D or that
it’s in the same direction as D. tD=<at,bt,ct>
Let P = (d, e, f) be a D = <a, b, c> y
point thespace,
then in line containing P,
in the direction of D consists x
of points P + tD = (d + at, e + bt, f + ct ).
Hence x(t) = at + d, y(t) = bt + e, z(t) = ct + f,
where t is any number, is a set of parametric
equations representing the line through
P = (d, e, f) in the direction D = <a, b, c>.
38. Parametric Equations of Lines
Example B. Find a set of parametric equations for the
line in the same direction as QR where Q = (3, –2, 5),
R = (1, 2, 1) that passes through P = (4, 5, 6).
39. Parametric Equations of Lines
Example B. Find a set of parametric equations for the
line in the same direction as QR where Q = (3, –2, 5),
R = (1, 2, 1) that passes through P = (4, 5, 6).
z
Q(3,–2,5) QR=D
R(1,2,1) y
x
40. Parametric Equations of Lines
Example B. Find a set of parametric equations for the
line in the same direction as QR where Q = (3, –2, 5),
R = (1, 2, 1) that passes through P = (4, 5, 6).
The vector QR = R – Q
= <1, 2, 1> – <3, –2, 5>
z
= <–2, 4, –4> = D is
a vector that gives the Q(3,–2,5) QR=D
direction of the line. R(1,2,1) y
x
41. Parametric Equations of Lines
Example B. Find a set of parametric equations for the
line in the same direction as QR where Q = (3, –2, 5),
R = (1, 2, 1) that passes through P = (4, 5, 6).
The vector QR = R – Q
= <1, 2, 1> – <3, –2, 5>
z
= <–2, 4, –4> = D is
a vector that gives the Q(3,–2,5) QR=D
direction of the line. So R(1,2,1) y
x(t) = –2t + 4
y(t) = 4t + 5 x
z(t) = –4t + 6
is the line through P that has the same direction as QR.
42. Parametric Equations of Lines
Example B. Find a set of parametric equations for the
line in the same direction as QR where Q = (3, –2, 5),
R = (1, 2, 1) that passes through P = (4, 5, 6).
The vector QR = R – Q
= <1, 2, 1> – <3, –2, 5> P(4,5,6)
z
= <–2, 4, –4> = D is P+t*QR
a vector that gives the Q(3,–2,5) QR=D
direction of the line. So R(1,2,1) y
x(t) = –2t + 4
y(t) = 4t + 5 x
z(t) = –4t + 6
is the line through P that has the same direction as QR.
43. Parametric Equations of Lines
Example B. Find a set of parametric equations for the
line in the same direction as QR where Q = (3, –2, 5),
R = (1, 2, 1) that passes through P = (4, 5, 6).
The vector QR = R – Q
= <1, 2, 1> – <3, –2, 5> P(4,5,6)
z
= <–2, 4, –4> = D is P+t*QR
a vector that gives the Q(3,–2,5) QR=D
direction of the line. So R(1,2,1) y
x(t) = –2t + 4
y(t) = 4t + 5 x
z(t) = –4t + 6
is the line through P that has the same direction as QR.
Definition: Two lines are parallel if they have the same
(or opposite) directional vector.
44. Parametric Equations of Lines
Let L represent the line in the last example, then we
may write L = P + t*D where D is a vector that gives
the direction of the line, P is a vector whose tip is on
the line L, and t as the variable. This is called the
vector-equation of the line.
P
z L = P+t*D
Q
QR = D
R
y
x
45. Parametric Equations of Lines
Let L represent the line in the last example, then we
may write L = P + t*D where D is a vector that gives
the direction of the line, P is a vector whose tip is on
the line L, and t as the variable. This is called the
vector-equation of the line.
P
Remark: We can't z L = P+t*D
represent a line in 3D with
Q
a single equation in the QR = D
variable x, y and z because R
y
the graph of such an
equation is a surface in 3D x
space in general.
46. Parametric Equations of Lines
Let L represent the line in the last example, then we
may write L = P + t*D where D is a vector that gives
the direction of the line, P is a vector whose tip is on
the line L, and t as the variable. This is called the
vector-equation of the line.
P
Remark: We can't z L = P+t*D
represent a line in 3D with
Q
a single equation in the QR = D
variable x, y and z because R
y
the graph of such an
equation is a surface in 3D x
space in general.
Another method of setting equations to represent a
line L is to give L as the intersection of two planes.
47. Parametric Equations of Lines
Let L be the line <t, t, t > where t is any real number,
as shown here. +
z
<1,1,1>
y
x
48. Parametric Equations of Lines
Let L be the line <t, t, t > where t is any real number,
as shown here. +
z
<1,1,1>
The parametric equations for L are
x(t) = t, y(t) = t, z(t) = t
y
x
49. Parametric Equations of Lines
Let L be the line <t, t, t > where t is any real number,
as shown here. +
z
<1,1,1>
The parametric equations for L are
x(t) = t, y(t) = t, z(t) = t
or that (t =) x = y = z. y
x
50. Parametric Equations of Lines
Let L be the line <t, t, t > where t is any real number,
as shown here. +
z
<1,1,1>
The parametric equations for L are
x(t) = t, y(t) = t, z(t) = t
or that (t =) x = y = z. y
These triple–equation is called the
x
symmetric equation of the line L.
51. Parametric Equations of Lines
Let L be the line <t, t, t > where t is any real number,
as shown here. +
z
<1,1,1>
The parametric equations for L are
x(t) = t, y(t) = t, z(t) = t
or that (t =) x = y = z. y
These triple–equation is called the
x
symmetric equation of the line L.
The symmetric equation actually consists of two
systems of linear equations,
52. Parametric Equations of Lines
Let L be the line <t, t, t > where t is any real number,
as shown here. +
z
<1,1,1>
The parametric equations for L are
x(t) = t, y(t) = t, z(t) = t
or that (t =) x = y = z. y
These triple–equation is called the
x
symmetric equation of the line L.
The symmetric equation actually consists of two
systems of linear equations, in this case
x=y x=y x=z
A: B: y = z C: y = z
x=z
53. Parametric Equations of Lines
Let L be the line <t, t, t > where t is any real number,
as shown here. +
z
<1,1,1>
The parametric equations for L are
x(t) = t, y(t) = t, z(t) = t
or that (t =) x = y = z. y
These triple–equation is called the
x
symmetric equation of the line L.
The symmetric equation actually consists of two
systems of linear equations, in this case
x=y x=y x=z
A: B: y = z C: y = z
x=z
Each system of equations consist of two planes and
L is the intersection of two planes.
