1. Vectors

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

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.

4. 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.

5. 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.
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.
u

6. 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.
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 B
with A as the base point and B as
the tip as AB . AB
u
A

7. 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.
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 B
with A as the base point and B as
the tip as AB . Many vector related AB
problems may be solved using the u
geometric diagrams based on these
A
arrows

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

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

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

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

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

13. Vectors
The length of a vector is called the magnitude of u or
the absolute value of u. It is denoted as |u| and |u| ≥ 0.
Vector Arithmetic
Equality of Vectors v
Two vectors u and v with the
same length and same direction u u=v
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. v 2v

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

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

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

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

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

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

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

21. Vector Addition
Vector Addition and The Parallelogram Rule
u
Geometrically, given two vectors u and v, v
the vector u + v is the diagonal vector of u+v
the parallelogram as shown, and u + v is v
u
also called the resultant of u and v in
The Parallelogram Rule
physics.
u
The Base to Tip Rule
Given two vectors u and v, u + v is the v
new vector formed by placing the base
The Base to Tip Rule
of one vector at the tip of the other.

22. Vector Addition
Vector Addition and The Parallelogram Rule
u
Geometrically, given two vectors u and v, v
the vector u + v is the diagonal vector of u+v
the parallelogram as shown, and u + v is v
u
also called the resultant of u and v in
The Parallelogram Rule
physics.
u
The Base to Tip Rule
v
Given two vectors u and v, u + v is the v
new vector formed by placing the base
The Base to Tip Rule
of one vector at the tip of the other.

23. Vector Addition
Vector Addition and The Parallelogram Rule
u
Geometrically, given two vectors u and v, v
the vector u + v is the diagonal vector of u+v
the parallelogram as shown, and u + v is v
u
also called the resultant of u and v in
The Parallelogram Rule
physics.
u
The Base to Tip Rule
v
Given two vectors u and v, u + v is the v u+v
new vector formed by placing the base
The Base to Tip Rule
of one vector at the tip of the other.

24. Vector Addition
Vector Addition and The Parallelogram Rule
u
Geometrically, given two vectors u and v, v
the vector u + v is the diagonal vector of u+v
the parallelogram as shown, and u + v is v
u
also called the resultant of u and v in
The Parallelogram Rule
physics.
u
The Base to Tip Rule
v
Given two vectors u and v, u + v is the v u+v
new vector formed by placing the base
The Base to Tip Rule
of one vector at the tip of the other.
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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

45. Vectors in a Coordinate System
A vector v placed in the coordinate
system with the base at the origin
(0, 0) is said to be in the standard
position.

46. Vectors in a Coordinate System
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
the standard position is (a, b), we write v as <a, b>.

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

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

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

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

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

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

53. Magnitudes of Vectors
The magnitude or the length of v = <a, b>
y
the vector v = <a, b> is given by
|v| = √a2 + b2
|v| = √a2 + b2 and the magnitude b
of |BT| = √Δx2 + Δy2. a
x

54. Magnitudes of Vectors
The magnitude or the length of v = <a, b>
y
the vector v = <a, b> is given by
|v| = √a2 + b2
|v| = √a2 + b2 and the magnitude b
of |BT| = √Δx2 + Δy2. a
x
Example B.
Find BT in the standard form |BT| if B = (1, –3) and
T = (5, 1). Draw.

55. Magnitudes of Vectors
The magnitude or the length of v = <a, b>
y
the vector v = <a, b> is given by
|v| = √a2 + b2
|v| = √a2 + b2 and the magnitude b
of |BT| = √Δx2 + Δy2. a
x
Example B.
Find BT in the standard form |BT| if B = (1, –3) and
T = (5, 1). Draw. y
BT=<4, 4>
T(5, 1)
x
B(1, –3)

56. Magnitudes of Vectors
The magnitude or the length of v = <a, b>
y
the vector v = <a, b> is given by
|v| = √a2 + b2
|v| = √a2 + b2 and the magnitude b
of |BT| = √Δx2 + Δy2. a
x
Example B.
Find BT in the standard form |BT| if B = (1, –3) and
T = (5, 1). Draw. y
BT=<4, 4>
BT = T – B = (5, 1) – (1, –3)
T(5, 1)
= <4, 4> x
B(1, –3)

57. Magnitudes of Vectors
The magnitude or the length of v = <a, b>
y
the vector v = <a, b> is given by
|v| = √a2 + b2
|v| = √a2 + b2 and the magnitude b
of |BT| = √Δx2 + Δy2. a
x
Example B.
Find BT in the standard form |BT| if B = (1, –3) and
T = (5, 1). Draw. y
BT=<4, 4>
BT = T – B = (5, 1) – (1, –3)
T(5, 1)
= <4, 4> and that x
|BT| = |<4, 4>| = √42 + 42 = √32 = 4√2 B(1, –3)

58. Magnitudes of Vectors
The magnitude or the length of v = <a, b>
y
the vector v = <a, b> is given by
|v| = √a2 + b2
|v| = √a2 + b2 and the magnitude b
of |BT| = √Δx2 + Δy2. a
x
Example B.
Find BT in the standard form |BT| if B = (1, –3) and
T = (5, 1). Draw. y
BT=<4, 4>
BT = T – B = (5, 1) – (1, –3)
T(5, 1)
= <4, 4> and that x
|BT| = |<4, 4>| = √42 + 42 = √32 = 4√2 B(1, –3)
Given vectors in the standard form <a, b>,
scalar-multiplication and vector addition are carried
out coordinate-wise, i.e. operations are done at each
coordinate.

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

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

61. Algebra of Standard Vectors
Example C. Let u = <–4, 1>, v = <3, –6>, find
a. 5u – 3v 5u – 3v = <–29, 23>
u
v

62. Algebra of Standard Vectors
Example C. Let u = <–4, 1>, v = <3, –6>, find
a. 5u – 3v 5u – 3v = <–29, 23>
5u
u
v

63. Algebra of Standard Vectors
Example C. Let u = <–4, 1>, v = <3, –6>, find
a. 5u – 3v 5u – 3v = <–29, 23>
5u –3v
u
v

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

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

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

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

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

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

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

71. Algebra of Standard Vectors
The Unit Coordinate Vectors

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

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

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

75. Algebra of Standard Vectors j = <0, 1>
The Unit Coordinate Vectors
A vector of length 1 is called a unit vector
so that in R2, the unit vectors are vectors i = <1, 0>
with tips on the unit circle.
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.

76. Algebra of Standard Vectors j = <0, 1>
The Unit Coordinate Vectors
A vector of length 1 is called a unit vector
so that in R2, the unit vectors are vectors i = <1, 0>
with tips on the unit circle.
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.

77. Algebra of Standard Vectors j = <0, 1>
The Unit Coordinate Vectors
A vector of length 1 is called a unit vector
so that in R2, the unit vectors are vectors i = <1, 0>
with tips on the unit circle.
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.
For example <3, 2> = 3i + 2j and that <–4, 1> = –4i +
1j.

78. Algebra of Standard Vectors j = <0, 1>
The Unit Coordinate Vectors
A vector of length 1 is called a unit vector
so that in R2, the unit vectors are vectors i = <1, 0>
with tips on the unit circle.
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.
For example <3, 2> = 3i + 2j and that <–4, 1> = –4i +
1j.
We may track vector calculation with i and j.

79. Algebra of Standard Vectors j = <0, 1>
The Unit Coordinate Vectors
A vector of length 1 is called a unit vector
so that in R2, the unit vectors are vectors i = <1, 0>
with tips on the unit circle.
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.
For example <3, 2> = 3i + 2j and that <–4, 1> = –4i +
1j.
We may track vector calculation with i and j.
For example <3, –4> – <1, –5> = (3i – 4j) – (i – 5j)

80. Algebra of Standard Vectors j = <0, 1>
The Unit Coordinate Vectors
A vector of length 1 is called a unit vector
so that in R2, the unit vectors are vectors i = <1, 0>
with tips on the unit circle.
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.
For example <3, 2> = 3i + 2j and that <–4, 1> = –4i +
1j.
We may track vector calculation with i and j.
For example <3, –4> – <1, –5> = (3i – 4j) – (i – 5j)
= 3i – 4j – i + 5j = 2i + j = <2, 1>.

81. Algebra of Standard Vectors j = <0, 1>
The Unit Coordinate Vectors
A vector of length 1 is called a unit vector
so that in R2, the unit vectors are vectors i = <1, 0>
with tips on the unit circle.
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.
For example <3, 2> = 3i + 2j and that <–4, 1> = –4i +
1j.
We may track vector calculation with i and j.
For example <3, –4> – <1, –5> = (3i – 4j) – (i – 5j)
= 3i – 4j – i + 5j = 2i + j = <2, 1>.
In many vector related calculations, it’s easier to keep

82. Algebra of Standard Vectors
All the above terminology and
operations may be applied to 3D
vectors.

83. Algebra of Standard Vectors
All the above terminology and
operations may be applied to 3D
vectors. A 3D vector in the
standard position is a vector who
is based at (0, 0, 0).

84. Algebra of Standard Vectors
All the above terminology and
operations may be applied to 3D
vectors. A 3D vector in the
standard position is a vector who
is based at (0, 0, 0).
We write v = <a, b, c> if the tip of
a 3D standard position vector v in
the is (a, b, c).

85. Algebra of Standard Vectors
All the above terminology and
operations may be applied to 3D z+
vectors. A 3D vector in the y
standard position is a vector who 1
2
is based at (0, 0, 0). x
–3
We write v = <a, b, c> if the tip of
a 3D standard position vector v in
u = <2,1,–3>
the is (a, b, c).

86. Algebra of Standard Vectors
All the above terminology and
operations may be applied to 3D z+
vectors. A 3D vector in the v = <–1,–2,1> y
standard position is a vector who 1
2
is based at (0, 0, 0). 1 –2
x
–3
We write v = <a, b, c> if the tip of –1
a 3D standard position vector v in
u = <2,1,–3>
the is (a, b, c).

87. Algebra of Standard Vectors
All the above terminology and
operations may be applied to 3D z+
vectors. A 3D vector in the v = <–1,–2,1> y
standard position is a vector who 1
2
is based at (0, 0, 0). 1 –2
x
–3
We write v = <a, b, c> if the tip of –1
a 3D standard position vector v in
u = <2,1,–3>
the is (a, b, c).
The zero vector is 0 = <0, 0, 0>.

88. Algebra of Standard Vectors
All the above terminology and
operations may be applied to 3D z +
vectors. A 3D vector in the v = <–1,–2,1> y
standard position is a vector who 1
2
is based at (0, 0, 0). 1 –2 x
–3
We write v = <a, b, c> if the tip of –1
a 3D standard position vector v in
u = <2,1,–3>
the is (a, b, c).
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>.

89. Algebra of Standard Vectors
All the above terminology and
operations may be applied to 3D z +
vectors. A 3D vector in the v = <–1,–2,1> y
standard position is a vector who 1
2
is based at (0, 0, 0). 1 –2 x
–3
We write v = <a, b, c> if the tip of –1
a 3D standard position vector v in
u = <2,1,–3>
the is (a, b, c).
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 2D
vectors.

90. Algebra of Standard Vectors
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.

91. Algebra of Standard Vectors
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.
Vector Addition (Parallelogram Rule)
Let u = <a, b, c>, v = <r, s, t>, then
u + v = <a + r, b + s, c + t>.

92. Algebra of Standard Vectors
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.
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 ( = √Δx2 + Δy2 + Δx2).

93. Algebra of Standard Vectors
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.
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 ( = √Δx2 + Δy2 + Δx2).
We write the coordinate unit vectors as
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>
<a, b, c> = ai + bj + ck

94. Algebra of Standard Vectors
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