vectors

1. Vectors
A vector, in 2 or higher dimensional space, is a
numerical measurement in a specified direction.

2. Vectors
A vector, in 2 or higher dimensional space, is a
numerical measurement in a specified direction.
We use the symbols u, v, and w (or u, v, and w)
to represent vectors in mathematics.

3. Vectors
A vector, in 2 or higher dimensional space, is a
numerical measurement in a specified direction.
We use the symbols u, v, and w (or u, v, and w)
to represent vectors in mathematics. In physics,
F, G, and H are used to represent force vectors.

4. Vectors
A vector, in 2 or higher dimensional space, is a
numerical measurement in a specified direction.
u
We use the symbols u, v, and w (or u, v, and w)
to represent vectors in mathematics. In physics,
F, G, and H are used to represent force vectors.
We may draw an arrow to represent a vector,
with the arrow pointing in the specified direction and
the length of the arrow indicating the numerical
measurement.

5. Vectors
A vector, in 2 or higher dimensional space, is a
numerical measurement in a specified direction.
u
A
B
AB
We use the symbols u, v, and w (or u, v, and w)
to represent vectors in mathematics. In physics,
F, G, and H are used to represent force vectors.
with A as the base point and B as
the tip as AB .
We may draw an arrow to represent a vector,
with the arrow pointing in the specified direction and
the length of the arrow indicating the numerical
measurement. We write the arrow or vector

6. Vectors
A vector, in 2 or higher dimensional space, is a
numerical measurement in a specified direction.
u
A
B
AB
We use the symbols u, v, and w (or u, v, and w)
to represent vectors in mathematics. In physics,
F, G, and H are used to represent force vectors.
with A as the base point and B as
the tip as AB . Many vector related
problems may be solved using the
geometric diagrams based on these
arrows.
We may draw an arrow to represent a vector,
with the arrow pointing in the specified direction and
the length of the arrow indicating the numerical
measurement. We write the arrow or vector

7. Vectors
The length of a vector u is called the magnitude of u or
the absolute value of u. It is denoted as |u| and |u| ≥ 0.

8. Vectors
Vector Arithmetic
The length of a vector u is called the magnitude of u or
the absolute value of u. It is denoted as |u| and |u| ≥ 0.

9. Vectors
Vector Arithmetic
Equality of Vectors
Two vectors u and v with the
same length and same direction
are equal, i.e. u = v.
The length of a vector u is called the magnitude of u or
the absolute value of u. It is denoted as |u| and |u| ≥ 0.

10. Vectors
Vector Arithmetic
Equality of Vectors
Two vectors u and v with the
same length and same direction
are equal, i.e. u = v.
u
v
u = v
The length of a vector u is called the magnitude of u or
the absolute value of u. It is denoted as |u| and |u| ≥ 0.

11. Vectors
Vector Arithmetic
Equality of Vectors
Two vectors u and v with the
same length and same direction
are equal, i.e. u = v.
Scalar Multiplication
Given a number λ and a vector v, λv
is the extension or the compression
of the vector v by a factor λ.
u
v
u = v
The length of a vector u is called the magnitude of u or
the absolute value of u. It is denoted as |u| and |u| ≥ 0.

12. Vectors
Vector Arithmetic
Equality of Vectors
Two vectors u and v with the
same length and same direction
are equal, i.e. u = v.
Scalar Multiplication
Given a number λ and a vector v, λv
is the extension or the compression
of the vector v by a factor λ.
u
v
v 2v
u = v
The length of a vector u is called the magnitude of u or
the absolute value of u. It is denoted as |u| and |u| ≥ 0.

13. Vectors
Vector Arithmetic
Equality of Vectors
Two vectors u and v with the
same length and same direction
are equal, i.e. u = v.
Scalar Multiplication
Given a number λ and a vector v, λv
is the extension or the compression
of the vector v by a factor λ.
If λ < 0, λv is in the opposite
direction of v.
u
v
v 2v
–2v –v
u = v
The length of a vector u is called the magnitude of u or
the absolute value of u. It is denoted as |u| and |u| ≥ 0.

14. Vectors
Vector Arithmetic
Equality of Vectors
Two vectors u and v with the
same length and same direction
are equal, i.e. u = v.
Scalar Multiplication
Given a number λ and a vector v, λv
is the extension or the compression
of the vector v by a factor λ.
If λ < 0, λv is in the opposite
direction of v.
u
v
v 2v
–2v
Finally, 0v = 0 , the zero vector.
–v
u = v
The length of a vector u is called the magnitude of u or
the absolute value of u. It is denoted as |u| and |u| ≥ 0.

15. Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown,
Vector Addition
Vector Addition and The Parallelogram Rule

16. u
v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown,
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule

17. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown,
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule

18. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule

19. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule

20. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
u
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
The Base to Tip Rule

21. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
v
u
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
The Base to Tip Rule

22. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
v
u
u + v
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
The Base to Tip Rule

23. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
v
u
u + v
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
It is easier to use the base-to-tip rule
to see that the order of the addition
does not matter for the sum of
three or more vectors.
The Base to Tip Rule

24. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
v
u
u + v
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
It is easier to use the base-to-tip rule
to see that the order of the addition
does not matter for the sum of
three or more vectors.
The Base to Tip Rule
u
v w

25. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
v
u
u + v
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
It is easier to use the base-to-tip rule
to see that the order of the addition
does not matter for the sum of
three or more vectors. u
The Base to Tip Rule
u
v w

26. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
v
u
u + v
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
It is easier to use the base-to-tip rule
to see that the order of the addition
does not matter for the sum of
three or more vectors. u
v
The Base to Tip Rule
u
v w

27. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
v
u
u + v
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
It is easier to use the base-to-tip rule
to see that the order of the addition
does not matter for the sum of
three or more vectors. u
v
w
u + v + w
The Base to Tip Rule
u
v w

28. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
v
u
u + v
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
It is easier to use the base-to-tip rule
to see that the order of the addition
does not matter for the sum of
three or more vectors. u
v
w
The Base to Tip Rule
u
v w

29. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
v
u
u + v
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
It is easier to use the base-to-tip rule
to see that the order of the addition
does not matter for the sum of
three or more vectors. u
v
w
u + v + w
The Base to Tip Rule
u
v w

30. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
v
u
u + v
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
It is easier to use the base-to-tip rule
to see that the order of the addition
does not matter for the sum of
three or more vectors. u
v
w
u + v + w
The Base to Tip Rule
w
u
v w

31. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
v
u
u + v
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
It is easier to use the base-to-tip rule
to see that the order of the addition
does not matter for the sum of
three or more vectors. u
v
w
u + v + w
The Base to Tip Rule
u w
u
v w

32. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
v
u
u + v
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
It is easier to use the base-to-tip rule
to see that the order of the addition
does not matter for the sum of
three or more vectors. u
v
w
u + v + w
The Base to Tip Rule
u
v
w
u
v w

33. u
v
u
v
u + v
Geometrically, given two vectors u and v,
the vector u + v is the diagonal vector of
the parallelogram as shown, and u + v is
also called the resultant of u and v in
physics.
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.
v
u
u + v
Vector Addition
Vector Addition and The Parallelogram Rule
The Parallelogram Rule
v
It is easier to use the base-to-tip rule
to see that the order of the addition
does not matter for the sum of
three or more vectors. u
v
w
u + v + w
The Base to Tip Rule
u
v
w
w + u + v
u
v w
=

34. The Tip to Tip Rule
Vector Subtraction
Given two vectors u and v, the vector
u – v = u +(–v) is the vector from the tip
of v to tip of u.
u
v

35. The Tip to Tip Rule u – v
Vector Subtraction
Given two vectors u and v, the vector
u – v = u +(–v) is the vector from the tip
of v to tip of u.
u
v

36. The Tip to Tip Rule u – v
Vector Subtraction
Given two vectors u and v, the vector
u – v = u +(–v) is the vector from the tip
of v to tip of u.
u
v
u v
w
Example A. Given the following three vectors
u, v and w, draw u – v – w.

37. The Tip to Tip Rule u – v
Vector Subtraction
Given two vectors u and v, the vector
u – v = u +(–v) is the vector from the tip
of v to tip of u.
u
v
u v
w
Example A. Given the following three vectors
u, v and w, draw u – v – w.
The vector u – v is
v
u

38. The Tip to Tip Rule u – v
Vector Subtraction
Given two vectors u and v, the vector
u – v = u +(–v) is the vector from the tip
of v to tip of u.
u
v
u v
w
Example A. Given the following three vectors
u, v and w, draw u – v – w.
The vector u – v is
v
u – v
u

39. The Tip to Tip Rule u – v
Vector Subtraction
Given two vectors u and v, the vector
u – v = u +(–v) is the vector from the tip
of v to tip of u.
u
v
u v
w
Example A. Given the following three vectors
u, v and w, draw u – v – w.
The vector u – v is
v
u – v
u
Hence the vector u – v – w is

40. The Tip to Tip Rule u – v
Vector Subtraction
Given two vectors u and v, the vector
u – v = u +(–v) is the vector from the tip
of v to tip of u.
u
v
u v
w
Example A. Given the following three vectors
u, v and w, draw u – v – w.
The vector u – v is
v
u – v
u
Hence the vector u – v – w is
u – v

41. The Tip to Tip Rule u – v
Vector Subtraction
Given two vectors u and v, the vector
u – v = u +(–v) is the vector from the tip
of v to tip of u.
u
v
u v
w
Example A. Given the following three vectors
u, v and w, draw u – v – w.
The vector u – v is
v
u – v
u
Hence the vector u – v – w is
u – v
w

42. The Tip to Tip Rule u – v
Vector Subtraction
Given two vectors u and v, the vector
u – v = u +(–v) is the vector from the tip
of v to tip of u.
u
v
u v
w
Example A. Given the following three vectors
u, v and w, draw u – v – w.
The vector u – v is
v
u – v
u
Hence the vector u – v – w is
u – v
w
u – v – w

43. The Tip to Tip Rule u – v
Vector Subtraction
Given two vectors u and v, the vector
u – v = u +(–v) is the vector from the tip
of v to tip of u.
u
v
u v
w
Example A. Given the following three vectors
u, v and w, draw u – v – w.
The vector u – v is
v
u – v
u
Hence the vector u – v – w is
u – v
w
u – v – w
Your turn: Draw u – (v + w). Show that it’s the same
as u – v – w.

44. Vectors
7,8,45,46,49,50,51.

45. A vector v placed in the coordinate
system with the base at the origin
(0, 0) is said to be in the standard
position. If the tip of the vector v in
Vectors in a Coordinate System
u = <3, 4>
v = <–1, 2>
x
y

46. A vector v placed in the coordinate
system with the base at the origin
(0, 0) is said to be in the standard
position. If the tip of the vector v in
Vectors in a Coordinate System
the standard position is (a, b), we write v in standard
form as <a, b>.

47. A vector v placed in the coordinate
system with the base at the origin
(0, 0) is said to be in the standard
position. If the tip of the vector v in
Vectors in a Coordinate System
u = <3, 4>
v = <–1, 2>
x
y
the standard position is (a, b), we write v in standard
form as <a, b>.

48. A vector v placed in the coordinate
system with the base at the origin
(0, 0) is said to be in the standard
position. If the tip of the vector v in
Vectors in a Coordinate System
u = <3, 4>
v = <–1, 2>
x
y
the standard position is (a, b), we write v in standard
form as <a, b>. The zero vector is 0 = <0, 0>.

49. A vector v placed in the coordinate
system with the base at the origin
(0, 0) is said to be in the standard
position. If the tip of the vector v in
Vectors in a Coordinate System
u = <3, 4>
v = <–1, 2>
x
y
the standard position is (a, b), we write v in standard
form as <a, b>. The zero vector is 0 = <0, 0>.
The vector BT with B = (a, b) at the base and
T = (c, d) the tip is BT = T – B = <c – a, d – b>.

50. A vector v placed in the coordinate
system with the base at the origin
(0, 0) is said to be in the standard
position. If the tip of the vector v in
Vectors in a Coordinate System
u = <3, 4>
v = <–1, 2>
x
y
the standard position is (a, b), we write v in standard
form as <a, b>. The zero vector is 0 = <0, 0>.
The vector BT with B = (a, b) at the base and
T = (c, d) the tip is BT = T – B = <c – a, d – b>.
Hence the vector from
B = (3, –1), to T = (–2, 4) is
BT = T – B =
B = (3, –1)
T = (–2, 4)
BT

51. A vector v placed in the coordinate
system with the base at the origin
(0, 0) is said to be in the standard
position. If the tip of the vector v in
Vectors in a Coordinate System
u = <3, 4>
v = <–1, 2>
x
y
the standard position is (a, b), we write v in standard
form as <a, b>. The zero vector is 0 = <0, 0>.
The vector BT with B = (a, b) at the base and
T = (c, d) the tip is BT = T – B = <c – a, d – b>.
Hence the vector from
B = (3, –1), to T = (–2, 4) is
BT = T – B = (–2, 4) – (3, –1)
=<–2 – 3, 4 – (–1) >
=<–5, 5 >
in the standard form.
B = (3, –1)
T = (–2, 4)
BT

52. A vector v placed in the coordinate
system with the base at the origin
(0, 0) is said to be in the standard
position. If the tip of the vector v in
Vectors in a Coordinate System
u = <3, 4>
v = <–1, 2>
x
y
the standard position is (a, b), we write v in standard
form as <a, b>. The zero vector is 0 = <0, 0>.
The vector BT with B = (a, b) at the base and
T = (c, d) the tip is BT = T – B = <c – a, d – b>.
Hence the vector from
B = (3, –1), to T = (–2, 4) is
BT = T – B = (–2, 4) – (3, –1)
=<–2 – 3, 4 – (–1) >
=<–5, 5 >
in the standard form.
BT in the standard
form based at (0, 0)
B = (3, –1)
T = (–2, 4)
BT = < –5, 5 > BT

53. The magnitude or the length of
the vector v = <a, b> is given by
|v| = a2 + b2
Magnitudes of Vectors
v = <a, b>
x
y
|v| = a2 + b2

54. The magnitude or the length of
the vector v = <a, b> is given by
|v| = a2 + b2
Example B.
Find BT in the standard form and |BT| if B = (1, –3)
and T = (5, 1). Draw.
Magnitudes of Vectors
v = <a, b>
x
y
|v| = a2 + b2

55. The magnitude or the length of
the vector v = <a, b> is given by
|v| = a2 + b2
Example B.
Find BT in the standard form and |BT| if B = (1, –3)
and T = (5, 1). Draw.
Magnitudes of Vectors
v = <a, b>
x
y
|v| = a2 + b2
BT = T – B = (5, 1) – (1, –3)
= <4, 4>
T(5, 1)
B(1, –3)
x
y

56. The magnitude or the length of
the vector v = <a, b> is given by
|v| = a2 + b2
Example B.
Find BT in the standard form and |BT| if B = (1, –3)
and T = (5, 1). Draw.
Magnitudes of Vectors
v = <a, b>
x
y
|v| = a2 + b2
BT = T – B = (5, 1) – (1, –3)
= <4, 4>
T(5, 1)
B(1, –3)
BT=<4, 4>
x
y

57. The magnitude or the length of
the vector v = <a, b> is given by
|v| = a2 + b2
Example B.
Find BT in the standard form and |BT| if B = (1, –3)
and T = (5, 1). Draw.
Magnitudes of Vectors
v = <a, b>
x
y
|v| = a2 + b2
BT = T – B = (5, 1) – (1, –3)
= <4, 4> and that
|BT| = |<4, 4>| = 42 + 42 = 32
T(5, 1)
BT=<4, 4>
x
y
B(1, –3)

58. The magnitude or the length of
the vector v = <a, b> is given by
|v| = a2 + b2
Example B.
Find BT in the standard form and |BT| if B = (1, –3)
and T = (5, 1). Draw.
Magnitudes of Vectors
v = <a, b>
x
y
|v| = a2 + b2
BT = T – B = (5, 1) – (1, –3)
= <4, 4> and that
|BT| = |<4, 4>| = 42 + 42 = 32
T(5, 1)
B(1, –3)
BT=<4, 4>
Given vectors in the standard form <a, b>,
scalar-multiplication and vector addition are carried
out coordinate-wise, i.e. operations are done on each
coordinate.
x
y

59. Scalar Multiplication
Let u = <a, b> and λ be a real number,
then λu = λ<a, b> = <λa, λb>.
Hence if u = <–2, 1>
then 2u = 2<–2, 1> = <–4, 2>
Algebra of Standard Vectors

60. Scalar Multiplication
Let u = <a, b> and λ be a real number,
then λu = λ<a, b> = <λa, λb>.
Hence if u = <–2, 1>
then 2u = 2<–2, 1> = <–4, 2>
Algebra of Standard Vectors
u=<–2, 1>
2u=2<–2, 1>
=<–4, 2>
x
y

61. Scalar Multiplication
Let u = <a, b> and λ be a real number,
then λu = λ<a, b> = <λa, λb>.
Hence if u = <–2, 1>
then 2u = 2<–2, 1> = <–4, 2>
Algebra of Standard Vectors
u=<–2, 1>
2u=2<–2, 1>
=<–4, 2>
x
y

62. Scalar Multiplication
Let u = <a, b> and λ be a real number,
then λu = λ<a, b> = <λa, λb>.
Hence if u = <–2, 1>
then 2u = 2<–2, 1> = <–4, 2>
(This corresponds to stretching
u to twice its length.)
Algebra of Standard Vectors
u=<–2, 1>
2u=2<–2, 1>
=<–4, 2>
x
y
Vector Addition
Let u = <a, b>, v = <c, d>
then u + v = <a + c, b + d>.

63. Scalar Multiplication
Let u = <a, b> and λ be a real number,
then λu = λ<a, b> = <λa, λb>.
Hence if u = <–2, 1>
then 2u = 2<–2, 1> = <–4, 2>
(This corresponds to stretching
u to twice its length.)
Algebra of Standard Vectors
u=<–2, 1>
2u=2<–2, 1>
=<–4, 2>
x
y
Vector Addition
Let u = <a, b>, v = <c, d>
then u + v = <a + c, b + d>.
Hence if u = <4, 1> and v = <–2, 3>
then u + v = <4, 1> + <–2, 3> = <2, 4>.

64. Scalar Multiplication
Let u = <a, b> and λ be a real number,
then λu = λ<a, b> = <λa, λb>.
Hence if u = <–2, 1>
then 2u = 2<–2, 1> = <–4, 2>
(This corresponds to stretching
u to twice its length.)
Algebra of Standard Vectors
u=<–2, 1>
2u=2<–2, 1>
=<–4, 2>
x
y
Vector Addition
Let u = <a, b>, v = <c, d>
then u + v = <a + c, b + d>.
x
Hence if u = <4, 1> and v = <–2, 3>
then u + v = <4, 1> + <–2, 3> = <2, 4>.
v=<–2, 3>

65. Scalar Multiplication
Let u = <a, b> and λ be a real number,
then λu = λ<a, b> = <λa, λb>.
Hence if u = <–2, 1>
then 2u = 2<–2, 1> = <–4, 2>
(This corresponds to stretching
u to twice its length.)
Algebra of Standard Vectors
u=<–2, 1>
2u=2<–2, 1>
=<–4, 2>
x
y
Vector Addition
Let u = <a, b>, v = <c, d>
then u + v = <a + c, b + d>.
u
v
u + v
x
v=<–2, 3>
u +v=<2, 4>
Hence if u = <4, 1> and v = <–2, 3>
then u + v = <4, 1> + <–2, 3> = <2, 4>.

66. Scalar Multiplication
Let u = <a, b> and λ be a real number,
then λu = λ<a, b> = <λa, λb>.
Hence if u = <–2, 1>
then 2u = 2<–2, 1> = <–4, 2>
(This corresponds to stretching
u to twice its length.)
Algebra of Standard Vectors
u=<–2, 1>
2u=2<–2, 1>
=<–4, 2>
x
y
Vector Addition
Let u = <a, b>, v = <c, d>
then u + v = <a + c, b + d>.
This is the same as the Parallelogram Rule.
u
v
u + v
x
v=<–2, 3>
u +v=<2, 4>
Hence if u = <4, 1> and v = <–2, 3>
then u + v = <4, 1> + <–2, 3> = <2, 4>.

67. Scalar Multiplication
Let u = <a, b> and λ be a real number,
then λu = λ<a, b> = <λa, λb>.
Hence if u = <–2, 1>
then 2u = 2<–2, 1> = <–4, 2>
(This corresponds to stretching
u to twice its length.)
Algebra of Standard Vectors
u=<–2, 1>
2u=2<–2, 1>
=<–4, 2>
x
y
Vector Addition
Let u = <a, b>, v = <c, d>
then u + v = <a + c, b + d>.
This is the same as the Parallelogram Rule.
u
v
u + v
x
v=<–2, 3>
u +v=<2, 4>
Hence if u = <4, 1> and v = <–2, 3>
then u + v = <4, 1> + <–2, 3> = <2, 4>.
Finally, we define u – v = u + (–v)

68. Example C. Let u= <–4, 1>, v= <3, –6>, find
a. 5u – 3v
Algebra of Standard Vectors
b. |5u – 3v|

69. Example C. Let u= <–4, 1>, v= <3, –6>, find
a. 5u – 3v
v
u
Algebra of Standard Vectors
b. |5u – 3v|

70. Example C. Let u= <–4, 1>, v= <3, –6>, find
a. 5u – 3v
= 5<–4, 1> – 3<3, –6>
= <–20, 5> + <–9, 18>
= <–29, 23>
v
u
Algebra of Standard Vectors
b. |5u – 3v|

71. Example C. Let u= <–4, 1>, v= <3, –6>, find
a. 5u – 3v
= 5<–4, 1> – 3<3, –6>
= <–20, 5> + <–9, 18>
= <–29, 23>
v
u
5u
Algebra of Standard Vectors
b. |5u – 3v|

72. Example C. Let u= <–4, 1>, v= <3, –6>, find
a. 5u – 3v
= 5<–4, 1> – 3<3, –6>
= <–20, 5> + <–9, 18>
= <–29, 23>
v
u
5u –3v
Your Turn
Given the u and v above, find 3u – 5v and |3u – 5v|.
Sketch the vectors.
Ans: 3u – 5v = <-27, 33> and |3u – 5v|  42.6
Algebra of Standard Vectors
b. |5u – 3v|

73. Example C. Let u= <–4, 1>, v= <3, –6>, find
a. 5u – 3v
= 5<–4, 1> – 3<3, –6>
= <–20, 5> + <–9, 18>
= <–29, 23>
v
u
5u –3v
5u – 3v = <–29, 23>
Algebra of Standard Vectors
b. |5u – 3v|

74. Example C. Let u= <–4, 1>, v= <3, –6>, find
a. 5u – 3v
= 5<–4, 1> – 3<3, –6>
= <–20, 5> + <–9, 18>
= <–29, 23>
v
u
5u –3v
5u – 3v = <–29, 23>
Algebra of Standard Vectors
b. |5u – 3v|
= (–29)2 + (23)2
= 1370  37.0

75. Example C. Let u= <–4, 1>, v= <3, –6>, find
a. 5u – 3v
= 5<–4, 1> – 3<3, –6>
= <–20, 5> + <–9, 18>
= <–29, 23>
v
u
5u –3v
5u – 3v = <–29, 23>
Your Turn
Given the u and v above, find 3u – 5v and |3u – 5v|.
Sketch the vectors.
Algebra of Standard Vectors
b. |5u – 3v|
= (–29)2 + (23)2
= 1370  37.0

76. Example C. Let u= <–4, 1>, v= <3, –6>, find
a. 5u – 3v
= 5<–4, 1> – 3<3, –6>
= <–20, 5> + <–9, 18>
= <–29, 23>
v
u
5u –3v
5u – 3v = <–29, 23>
Your Turn
Given the u and v above, find 3u – 5v and |3u – 5v|.
Sketch the vectors.
Ans: 3u – 5v = <-27, 33> and |3u – 5v|  42.6
Algebra of Standard Vectors
b. |5u – 3v|
= (–29)2 + (23)2
= 1370  37.0

77. A vector of length 1 is called a unit vector.
Algebra of Standard Vectors
The Unit Coordinate Vectors

78. A vector of length 1 is called a unit vector.
Hence in R2, the unit vectors are vectors
with tips on the unit circle.
Algebra of Standard Vectors
The Unit Coordinate Vectors

79. A vector of length 1 is called a unit vector.
Hence in R2, the unit vectors are vectors
with tips on the unit circle.
Algebra of Standard Vectors
The Unit Coordinate Vectors
Let i = <1, 0> and j = <0, 1> be the unit vectors along
the positive coordinate axes as shown.
i = <1, 0>
j = <0, 1>

80. A vector of length 1 is called a unit vector.
Hence in R2, the unit vectors are vectors
with tips on the unit circle.
Algebra of Standard Vectors
The Unit Coordinate Vectors
Let i = <1, 0> and j = <0, 1> be the unit vectors along
the positive coordinate axes as shown.
Then a vector <a, b> may be written as
<a, b> = a<1, 0> + b<0, 1> = ai + bj.
i = <1, 0>
j = <0, 1>

81. A vector of length 1 is called a unit vector.
Hence in R2, the unit vectors are vectors
with tips on the unit circle.
Algebra of Standard Vectors
The Unit Coordinate Vectors
Let i = <1, 0> and j = <0, 1> be the unit vectors along
the positive coordinate axes as shown.
Then a vector <a, b> may be written as
<a, b> = a<1, 0> + b<0, 1> = ai + bj.
u = <a, b>
i = <1, 0>
j = <0, 1>
i
j

82. A vector of length 1 is called a unit vector.
Hence in R2, the unit vectors are vectors
with tips on the unit circle.
Algebra of Standard Vectors
The Unit Coordinate Vectors
Let i = <1, 0> and j = <0, 1> be the unit vectors along
the positive coordinate axes as shown.
Then a vector <a, b> may be written as
<a, b> = a<1, 0> + b<0, 1> = ai + bj.
u = <a, b>
= ai + bj
ai
bj
i = <1, 0>
j = <0, 1>
i
j

83. A vector of length 1 is called a unit vector.
Hence in R2, the unit vectors are vectors
with tips on the unit circle.
For example <3, 2> = 3i + 2j and that <–4, 1> = –4i + j.
Algebra of Standard Vectors
The Unit Coordinate Vectors
Let i = <1, 0> and j = <0, 1> be the unit vectors along
the positive coordinate axes as shown.
Then a vector <a, b> may be written as
<a, b> = a<1, 0> + b<0, 1> = ai + bj.
u = <a, b>
= ai + bj
ai
bj
i = <1, 0>
j = <0, 1>
i
j

84. A vector of length 1 is called a unit vector.
Hence in R2, the unit vectors are vectors
with tips on the unit circle.
For example <3, 2> = 3i + 2j and that <–4, 1> = –4i + j.
We may perform vector calculations with i and j
as variables so <3, 2> – <–4, 1> = 3i + 2j – (–4i + j)
= 7i + j = <2, 1>.
Algebra of Standard Vectors
The Unit Coordinate Vectors
Let i = <1, 0> and j = <0, 1> be the unit vectors along
the positive coordinate axes as shown.
Then a vector <a, b> may be written as
<a, b> = a<1, 0> + b<0, 1> = ai + bj.
u = <a, b>
= ai + bj
ai
bj
i = <1, 0>
j = <0, 1>
i
j

85. A vector of length 1 is called a unit vector.
Hence in R2, the unit vectors are vectors
with tips on the unit circle.
For example <3, 2> = 3i + 2j and that <–4, 1> = –4i + j.
We may perform vector calculations with i and j
as variables so <3, 2> – <–4, 1> = 3i + 2j – (–4i + j)
= 7i + j = <2, 1>.
In many vector related calculations, it’s easier to keep
track of the algebra with i and j.
Algebra of Standard Vectors
The Unit Coordinate Vectors
Let i = <1, 0> and j = <0, 1> be the unit vectors along
the positive coordinate axes as shown.
Then a vector <a, b> may be written as
<a, b> = a<1, 0> + b<0, 1> = ai + bj.
u = <a, b>
= ai + bj
ai
bj
i = <1, 0>
j = <0, 1>
i
j

86. All the above terminology and
operations may be applied to 3D
vectors. A 3D vector in the
standard position is a vector that
is based at (0, 0, 0).
We write v = <a, b, c> if the tip of
a 3D standard position vector v is
(a, b, c).
Algebra of Standard Vectors
The zero vector is 0 = <0, 0, 0>.
If A = <a, b, c>, B = <r, s, t>, then the vector AB in
the standard position is B – A = <r – a, s – b, t – c>.
Scalar-multiplication, vector addition, and the norm
(magnitude) are defined similarly as in the case
with 2D vectors.

87. 3D Scalar-Multiplication
Let u = <a, b, c> and λ be a real number, then
λu = λ<a, b, c> = <λa, λb, λc>.
This corresponds to stretching u by a factor of λ.
Algebra of Standard Vectors
3D Vector Addition (Parallelogram Rule)
Let u = <a, b, c>, v = <r, s, t>, then
u + v = <a + r, b + s, c + t>.
The norm (magnitude or length) of u
is |u| = a2 + b2 + c2.
The unit-coordinate-vectors are
i = <1, 0, 0>, j = <0, 1, 0> and k = <0, 0, 1>,
then any vector <a, b, c> may be written as
<a, b, c> = a<1, 0, 0> + b<0, 1, 0> + c<0, 0, 1>,
so <a, b, c> = ai + bj + ck.

88. Example D. Let u = <–1, 1, 2>, v = <1, 3, 3>.
Find 3u – 2v, write 3u – 2v in terms of i , j, k, and
find |3u – 2v|.
3u – 2v = 3<–1, 1, 2> – 2<1, 3, 3>
= <–3, 3, 6> + <–2, –6, –6>
= <–5, –3, 0>
= –5i – 3j + 0k
In particular |3u – 2v| = 25 + 9 + 0 = 34
Algebra of Standard Vectors