vectors

1. Vectors

2. Vectors
A vector is a numerical measurement in a specified
direction in R2, R3 or higher dimensional spaces.
We draw arrows to represent vectors and use
vectors to track forces, movements or directional
changes. Many problems are solved using
geometric diagrams based on these arrows.

3. Vectors
A vector is a numerical measurement in a specified
direction in R2, R3 or higher dimensional spaces.
We draw arrows to represent vectors and use
vectors to track forces, movements or directional
changes. Many problems are solved using
geometric diagrams based on these arrows.
We write u, v, and w (or u, v, and w) to represent
vectors or AB where A is a specified base (point)
and the B is the tip of the arrow.

4. Vectors
A vector is a numerical measurement in a specified
direction in R2, R3 or higher dimensional spaces.
We draw arrows to represent vectors and use
vectors to track forces, movements or directional
changes. Many problems are solved using
geometric diagrams based on these arrows.
We write u, v, and w (or u, v, and w) to represent
vectors or AB where A is a specified base (point)
and the B is the tip of the arrow.
The length of a vector represents
the quantity and it is the magnitude
or the absolute value of u, and it is
denoted as |u|. Given u and v,
u ± v may be obtained geometrically.

5. Vector Arithmetic

6. Vector Arithmetic
Equality of vectors:
Two vectors u and v with
the same length and same
direction are equal, i.e. u=v.

7. Vector Arithmetic
Equality of vectors:
Two vectors u and v with
the same length and same
direction are equal, i.e. u=v.

8. 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:

9. 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.

10. 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.v

11. 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.v
2v

12. 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.v
2v
v
2
1

13. Vector Arithmetic
Equality of vectors:
Two vectors u and v with
the same length and same
direction are equal, i.e. u=v.
v
2v
v
2
1
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.

14. Vector Arithmetic
Equality of vectors:
Two vectors u and v with
the same length and same
direction are equal, i.e. u=v.
v
2v
v
2
1
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.-2v

15. Vector Arithmetic
Equality of vectors:
Two vectors u and v with
the same length and same
direction are equal, i.e. u=v.
v
2v
v
2
1
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.
0v = 0 , the zero vector.
-2v

16. Vector Arithmetic – Vector Addition

17. u
v
The Parallelogram Addition Rule
Given two vectors u and v,
u + v is the diagonal of
the parallelogram as shown.
Vector Arithmetic – Vector Addition

18. u
v
u
v
u + v
The Parallelogram Addition Rule
Given two vectors u and v,
u + v is the diagonal of
the parallelogram as shown.
u + v is also called the resultant of
u and v.
Vector Arithmetic – Vector Addition

19. u
u
v
u + v
The Parallelogram Addition Rule
Given two vectors u and v,
u + v is the diagonal of
the parallelogram as shown.
u + v is also called the resultant of
u and v.
The Base to Tip Addition 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
v
Vector Arithmetic – Vector Addition
v

20. u
v
u
v
u + v
The Parallelogram Addition Rule
Given two vectors u and v,
u + v is the diagonal of
the parallelogram as shown.
u + v is also called the resultant of
u and v.
The Base to Tip Addition 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
v
u
Vector Arithmetic – Vector Addition

21. u
v
u
v
u + v
The Parallelogram Addition Rule
Given two vectors u and v,
u + v is the diagonal of
the parallelogram as shown.
u + v is also called the resultant of
u and v.
The Base to Tip Addition 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
v
u
Vector Arithmetic – Vector Addition
u + v

22. It is easier to use the
base-to-tip rule to visualize
the sum of three or more
vectors. The order of the
addition does not matter.
Example A: Add by
placing the base of one
vector over the tip of
another.
Vector Arithmetic – Vector Addition

23. It is easier to use the
base-to-tip rule to visualize
the sum of three or more
vectors. The order of the
addition does not matter.
Example A: Add by
placing the base of one
vector over the tip of
another.
Vector Arithmetic – Vector Addition

24. It is easier to use the
base-to-tip rule to visualize
the sum of three or more
vectors. The order of the
addition does not matter.
Example A: Add by
placing the base of one
vector over the tip of
another.
Vector Arithmetic – Vector Addition

25. It is easier to use the
base-to-tip rule to visualize
the sum of three or more
vectors. The order of the
addition does not matter.
Example A: Add by
placing the base of one
vector over the tip of
another.
the sum
Vector Arithmetic – Vector Addition

26. It is easier to use the
base-to-tip rule to visualize
the sum of three or more
vectors. The order of the
addition does not matter.
Example A: Add by
placing the base of one
vector over the tip of
another.
the sum
Given two vectors u, v, and
the angle A between them,
to find |u+v| the magnitude
of their sum , use the
supplementary angle
180 – A and the Cosine
Law.
Vector Arithmetic – Vector Addition

27. Vector Arithmetic
Vector Subtraction

28. Vector Arithmetic
Vector Subtraction
The Tip to Tip rule:
u
v

29. Vector Arithmetic
Vector Subtraction
The Tip to Tip rule:
Given two vectors u and v,
u – v is the new vector formed
from the tip of v to tip of the u.
u
v

30. Vector Arithmetic
Vector Subtraction
The Tip to Tip rule:
Given two vectors u and v,
u – v is the new vector formed
from the tip of v to tip of the u.
u
v
u – v

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

32. 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>.
Vectors in a Coordinate System

33. >
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>.
Vectors in a Coordinate System

34. 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>.
The zero vector is 0 =<0, 0>
Vectors in a Coordinate System

35. 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>.
The zero vector is 0 =<0, 0>
A vector v may also specified by
it's starting point its base B
and it's tip T, v is also denoted
as BT.
Vectors in a Coordinate System

36. Given a vector v with base at
the point B=(a, b) and the tip
at the point T=(c, d), then BT
in the standard form is v where
v=BT=T – B= <c – a, d – b>
Vectors in a Coordinate System

37. Example A: Let B=(3, -1), and
T=(-2, 4). Find the vector BT
in the standard form.
Given a vector v with base at
the point B=(a, b) and the tip
at the point T=(c, d), then BT
in the standard form is v where
v=BT=T – B= <c – a, d – b>
Vectors in a Coordinate System

38. Example A: Let B=(3, -1), and
T=(-2, 4). Find the vector BT
in the standard form.
Given a vector v with base at
the point B=(a, b) and the tip
at the point T=(c, d), then BT
in the standard form is v where
v=BT=T – B= <c – a, d – b>
Vectors in a Coordinate System
B=(3, -1)
T=(-2, 4)
BT

39. Example A: Let B=(3, -1), and
T=(-2, 4). Find the vector BT
in the standard form.
BT=T – B
Given a vector v with base at
the point B=(a, b) and the tip
at the point T=(c, d), then BT
in the standard form is v where
v=BT=T – B= <c – a, d – b>
Vectors in a Coordinate System
B=(3, -1)
T=(-2, 4)
BT

40. Example A: Let B=(3, -1), and
T=(-2, 4). Find the vector BT
in the standard form.
BT=T – B = (-2, 4) – (3, -1)
Given a vector v with base at
the point B=(a, b) and the tip
at the point T=(c, d), then BT
in the standard form is v where
v=BT=T – B= <c – a, d – b>
Vectors in a Coordinate System
B=(3, -1)
T=(-2, 4)
BT

41. Example A: Let B=(3, -1), and
T=(-2, 4). Find the vector BT
in the standard form.
BT=T – B = (-2, 4) – (3, -1)
=< -2 – 3, 4 – (-1) >
Given a vector v with base at
the point B=(a, b) and the tip
at the point T=(c, d), then BT
in the standard form is v where
v=BT=T – B= <c – a, d – b>
Vectors in a Coordinate System
B=(3, -1)
T=(-2, 4)
BT

42. Example A: Let B=(3, -1), and
T=(-2, 4). Find the vector BT
in the standard form.
BT=T – B = (-2, 4) – (3, -1)
=< -2 – 3, 4 – (-1) >
=< -5, 5 > in the standard
form.
Given a vector v with base at
the point B=(a, b) and the tip
at the point T=(c, d), then BT
in the standard form is v where
v=BT=T – B= <c – a, d – b>
Vectors in a Coordinate System
B=(3, -1)
T=(-2, 4)
BT

43. Example A: Let B=(3, -1), and
T=(-2, 4). Find the vector BT
in the standard form.
BT=T – B = (-2, 4) – (3, -1)
=< -2 – 3, 4 – (-1) >
=< -5, 5 > in the standard
form.
Given a vector v with base at
the point B=(a, b) and the tip
at the point T=(c, d), then BT
in the standard form is v where
v=BT=T – B= <c – a, d – b>
BT in the standard
form based at (0, 0)
Vectors in a Coordinate System
B=(3, -1)
T=(-2, 4)
< -5, 5 >
BT

44. Vertical format:
( -2, 4 )
– ( 3, -1 )
Example A: Let B=(3, -1), and
T=(-2, 4). Find the vector BT
in the standard form.
BT=T – B = (-2, 4) – (3, -1)
=< -2 – 3, 4 – (-1) >
=< -5, 5 > in the standard
form.
Given a vector v with base at
the point B=(a, b) and the tip
at the point T=(c, d), then BT
in the standard form is v where
v=BT=T – B= <c – a, d – b>
BT in the standard
form based at (0, 0)
Vectors in a Coordinate System
B=(3, -1)
T=(-2, 4)
< -5, 5 >
BT

45. Vertical format:
( -2, 4 )
– ( 3, -1 )
<-5, 5>
Example A: Let B=(3, -1), and
T=(-2, 4). Find the vector BT
in the standard form.
BT=T – B = (-2, 4) – (3, -1)
=< -2 – 3, 4 – (-1) >
=< -5, 5 > in the standard
form.
Given a vector v with base at
the point B=(a, b) and the tip
at the point T=(c, d), then BT
in the standard form is v where
v=BT=T – B= <c – a, d – b>
BT in the standard
form based at (0, 0)
Vectors in a Coordinate System
B=(3, -1)
T=(-2, 4)
< -5, 5 >
BT

46. The magnitude or the
length of the vector
v = <a, b> is given by
|v| = a2 + b2
Magnitude of a Vector

47. The magnitude or the
length of the vector
v = <a, b> is given by
|v| = a2 + b2
Magnitude of a Vector

48. The magnitude or the
length of the vector
v = <a, b> is given by
|v| = a2 + b2
Magnitude of a Vector
Example B: Find |BT| if
B = (1, -3) and T = (5, 1).

49. The magnitude or the
length of the vector
v = <a, b> is given by
|v| = a2 + b2
Magnitude of a Vector
Example B: Find |BT| if
B = (1, -3) and T = (5, 1).
We need v = BT = T – B
in the standard form.

50. The magnitude or the
length of the vector
v = <a, b> is given by
|v| = a2 + b2
Magnitude of a Vector
Example B: Find |BT| if
B = (1, -3) and T = (5, 1).
We need v = BT = T – B
in the standard form. Vertical format:
( 5, 1 )
– ( 1, -3)

51. The magnitude or the
length of the vector
v = <a, b> is given by
|v| = a2 + b2
Magnitude of a Vector
Example B: Find |BT| if
B = (1, -3) and T = (5, 1).
We need v = BT = T – B
in the standard form. Vertical format:
( 5, 1 )
– ( 1, -3)
v = ( 4, 4)

52. The magnitude or the
length of the vector
v = <a, b> is given by
|v| = a2 + b2
Magnitude of a Vector
Example B: Find |BT| if
B = (1, -3) and T = (5, 1).
We need v = BT = T – B
in the standard form. Vertical format:
( 5, 1 )
– ( 1, -3)
v = ( 4, 4)
So |v| = |<4, 4>|

53. The magnitude or the
length of the vector
v = <a, b> is given by
|v| = a2 + b2
Magnitude of a Vector
Example B: Find |BT| if
B = (1, -3) and T = (5, 1).
We need v = BT = T – B
in the standard form. Vertical format:
( 5, 1 )
– ( 1, -3)
v = ( 4, 4)
So |v| = |<4, 4>|
= 42 + 42
= 32

54. The magnitude or the
length of the vector
v = <a, b> is given by
|v| = a2 + b2
Magnitude of a Vector
Example B: Find |BT| if
B = (1, -3) and T = (5, 1).
We need v = BT = T – B
in the standard form. Vertical format:
( 5, 1 )
– ( 1, -3)
v = ( 4, 4)
So |v| = |<4, 4>|
= 42 + 42
= 32 = 42  5.64

55. Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Algebra of Standard Vectors

56. Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Scalar-multiplication:
Let u=<a, b> and λ be a
real number, then
λu= λ<a, b> = <λa, λb>.
Algebra of Standard Vectors

57. Example C:
If u = <2, -1> then 2u = 2<2, -1> = <4, -2>
Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Scalar-multiplication:
Let u=<a, b> and λ be a
real number, then
λu= λ<a, b> = <λa, λb>.
Algebra of Standard Vectors

58. Example C:
If u = <2, -1> then 2u = 2<2, -1> = <4, -2>
Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Scalar-multiplication:
Let u=<a, b> and λ be a
real number, then
λu= λ<a, b> = <λa, λb>.
Algebra of Standard Vectors

59. Example C:
If u = <2, -1> then 2u = 2<2, -1> = <4, -2>
(This corresponds to stretching u to twice its length.)
Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Scalar-multiplication:
Let u=<a, b> and λ be a
real number, then
λu= λ<a, b> = <λa, λb>.
Algebra of Standard Vectors

60. Example C:
If u = <2, -1> then 2u = 2<2, -1> = <4, -2>
(This corresponds to stretching u to twice its length.)
Algebra of Standard Vectors
Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Scalar-multiplication:
Let u=<a, b> and λ be a
real number, then
λu= λ<a, b> = <λa, λb>.

61. Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Vector Addition:
Let u=<a, b>, v=<c, d>
then u + v = <a+c, b+d>.
Algebra of Standard Vectors

62. Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Vector Addition:
Let u=<a, b>, v=<c, d>
then u + v = <a+c, b+d>.
Algebra of Standard Vectors
Example D: Let u = <2, -1>, v = <-3, -2>,
a. u + v =
v=<-3, -2>
u=<2, -1>

63. Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Vector Addition:
Let u=<a, b>, v=<c, d>
then u + v = <a+c, b+d>.
Algebra of Standard Vectors
Example D: Let u = <2, -1>, v = <-3, -2>,
a. u + v = <2+(-3), -1+(-2)> = <-1,-3>
v=<-3, -2>
u=<2, -1>

64. Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Vector Addition:
Let u=<a, b>, v=<c, d>
then u + v = <a+c, b+d>.
Algebra of Standard Vectors
Example D: Let u = <2, -1>, v = <-3, -2>,
a. u + v = <2+(-3), -1+(-2)> = <-1,-3>
v=<-3, -2>
u + v =<-1,-3>
u=<2, -1>

65. Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Vector Addition:
Let u=<a, b>, v=<c, d>
then u + v = <a+c, b+d>.
We define u – v = u + (- v)
Algebra of Standard Vectors
Example D: Let u = <2, -1>, v = <-3, -2>,
a. u + v = <2+(-3), -1+(-2)> = <-1,-3>
v=<-3, -2>
u + v =<-1,-3>
u=<2, -1>

66. Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Vector Addition:
Let u=<a, b>, v=<c, d>
then u + v = <a+c, b+d>.
We define u – v = u + (- v)
Algebra of Standard Vectors
Example D: Let u = <2, -1>, v = <-3, -2>,
a. u + v = <2+(-3), -1+(-2)> = <-1,-3>
v=<-3, -2>
b. u – v = <2 – (-3), -1 – (-2)> = <5, 1>
u + v =<-1,-3>
u=<2, -1>

67. Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Vector Addition:
Let u=<a, b>, v=<c, d>
then u + v = <a+c, b+d>.
We define u – v = u + (- v)
Algebra of Standard Vectors
Example D: Let u = <2, -1>, v = <-3, -2>,
a. u + v = <2+(-3), -1+(-2)> = <-1,-3>
u=<2, -1>
v=<-3, -2>
u + v =<-1,-3>
b. u – v = <2 – (-3), -1 – (-2)> = <5, 1>
-v=< 3, 2>

68. Scalar-multiplication and
vector addition are done
coordinate-wise, i.e.
operations are done at
each coordinate.
Vector Addition:
Let u=<a, b>, v=<c, d>
then u + v = <a+c, b+d>.
We define u – v = u + (- v)
Algebra of Standard Vectors
Example D: Let u = <2, -1>, v = <-3, -2>,
a. u + v = <2+(-3), -1+(-2)> = <-1,-3>
u=<2, -1>
v=<-3, -2>
u + v =<-1,-3>
b. u – v = <2 – (-3), -1 – (-2)> = <5, 1>
-v=< 3, 2>
u–v=<5,1>

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

70. Example E: Let u= <-4, 1>, v= <3, -6>, find
a. 5u – 3v
Algebra of Standard Vectors
b. |5u – 3v|
v
u

71. Example E: Let u= <-4, 1>, v= <3, -6>, find
a. 5u – 3v
Algebra of Standard Vectors
b. |5u – 3v|
v
u
5u

72. Example E: Let u= <-4, 1>, v= <3, -6>, find
a. 5u – 3v
Algebra of Standard Vectors
b. |5u – 3v|
v
u
5u -3v

73. Example E: Let u= <-4, 1>, v= <3, -6>, find
a. 5u – 3v
= 5<-4, 1> – 3<3, -6>
Algebra of Standard Vectors
b. |5u – 3v|
v
u
5u -3v

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

75. Example E: Let u= <-4, 1>, v= <3, -6>, find
a. 5u – 3v
= 5<-4, 1> – 3<3, -6>
= <-20, 5> + <-9, 18>
= <-29, 23>
Algebra of Standard Vectors
v
u
5u -3v
5u – 3v
Such a sum of multiples
of u and v is called
a linear combination
of u and v.
b. |5u – 3v|

76. Example E: Let u= <-4, 1>, v= <3, -6>, find
a. 5u – 3v
= 5<-4, 1> – 3<3, -6>
= <-20, 5> + <-9, 18>
= <-29, 23>
Algebra of Standard Vectors
v
u
5u -3v
5u – 3v
Such a sum of multiples
of u and v is called
a linear combination
of u and v.
b. |5u – 3v|
= (-29)2 + (23)2
= 1370  37.0

77. Algebra of Vectors in the
Standard Position
Let i = <1, 0>, j = <0, 1>,
then any vector <a, b> may
be written as
a<1, 0>+b<0, 1>= ai + bj
i = <1, 0>
j = <0, 1>

78. Algebra of Vectors in the
Standard Position
Let i = <1, 0>, j = <0, 1>,
then any vector <a, b> may
be written as
a<1, 0>+b<0, 1>= ai + bj
i = <1, 0>
j = <0, 1>
Example F:
A. <3, 2> = 3i + 2j

79. Algebra of Vectors in the
Standard Position
Let i = <1, 0>, j = <0, 1>,
then any vector <a, b> may
be written as
a<1, 0>+b<0, 1>= ai + bj
i = <1, 0>
j = <0, 1>
Example F:
A. <3, 2> = 3i + 2j
B. <-4, 1> = -4i + 1j

80. Algebra of Vectors in the
Standard Position
Let i = <1, 0>, j = <0, 1>,
then any vector <a, b> may
be written as
a<1, 0>+b<0, 1>= ai + bj
i = <1, 0>
j = <0, 1>
Example F:
A. <3, 2> = 3i + 2j
B. <-4, 1> = -4i + 1j
C. <3, -4> – <1, -5>
= (3i – 4j) – (i – 5j)

81. Algebra of Vectors in the
Standard Position
Let i = <1, 0>, j = <0, 1>,
then any vector <a, b> may
be written as
a<1, 0>+b<0, 1>= ai + bj
i = <1, 0>
j = <0, 1>
Example F:
A. <3, 2> = 3i + 2j
B. <-4, 1> = -4i + 1j
C. <3, -4> – <1, -5>
= (3i – 4j) – (i – 5j)
= 3i – 4j – i + 5j
= 2i + j = <2, 1>

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

83. All the above terminology and
operations may be applied to 3D
vectors.
A 3D vector v in the standard position
is an arrow with its base point
at (0, 0, 0).
Standard 3D Vectors

84. All the above terminology and
operations may be applied to 3D
vectors.
A 3D vector v in the standard position
is an arrow with its base point
at (0, 0, 0).
If the tip of a 3D standard position
vector v in the is (a, b, c), then we
write v = <a, b, c>.
Example G:
a. Draw the vectors
u=<2, 1, -3>
v=<-1, -2, 1>
x
y
z+
2
1
-3
u
Standard 3D Vectors
(2, 1, 0)

85. All the above terminology and
operations may be applied to 3D
vectors.
A 3D vector v in the standard position
is an arrow with its base point
at (0, 0, 0).
If the tip of a 3D standard position
vector v in the is (a, b, c), then we
write v = <a, b, c>.
x
y
z+
2
1
-3
u
-1
-2
1
v
Standard 3D Vectors
(2, 1, 0)
Example G:
a. Draw the vectors
u=<2, 1, -3>
v=<-1, -2, 1>

86. All the above terminology and
operations may be applied to 3D
vectors.
A 3D vector v in the standard position
is an arrow with its base point
at (0, 0, 0).
If the tip of a 3D standard position
vector v in the is (a, b, c), then we
write v = <a, b, c>.
The zero vector is 0 = <0, 0, 0>
x
y
z+
2
1
-3
u
-1
-2
1
v
Standard 3D Vectors
(2, 1, 0)
Example G:
a. Draw the vectors
u=<2, 1, -3>
v=<-1, -2, 1>

87. All the above terminology and
operations may be applied to 3D
vectors.
A 3D vector v in the standard position
is an arrow with its base point
at (0, 0, 0).
If the tip of a 3D standard position
vector v in the is (a, b, c), then we
write v = <a, b, c>.
The zero vector is 0 = <0, 0, 0>
Given two points
A=(a, b, c) and B=(r, s, t),
then the vector AB in the standard
position is given by
AB = B – A = <r – a, s – b, t – c>.
x
y
z+
2
1
-3
u
-1
-2
1
v
Standard 3D Vectors
(2, 1, 0)
Example G:
a. Draw the vectors
u=<2, 1, -3>
v=<-1, -2, 1>

88. All the above terminology and
operations may be applied to 3D
vectors.
A 3D vector v in the standard position
is an arrow with its base point
at (0, 0, 0).
If the tip of a 3D standard position
vector v in the is (a, b, c), then we
write v = <a, b, c>.
The zero vector is 0 = <0, 0, 0>
Given two points
A=(a, b, c) and B=(r, s, t),
then the vector AB in the standard
position is given by
AB = B – A = <r – a, s – b, t – c>.
x
y
z+
2
1
-3
u
-1
-2
1
v
Standard 3D Vectors
b. Find the vector AB if
A=(2, 1, -3) and
B=(-1, -2, 1).
(2, 1, 0)
Example G:
a. Draw the vectors
u=<2, 1, -3>
v=<-1, -2, 1>

89. All the above terminology and
operations may be applied to 3D
vectors.
A 3D vector v in the standard position
is an arrow with its base point
at (0, 0, 0).
If the tip of a 3D standard position
vector v in the is (a, b, c), then we
write v = <a, b, c>.
The zero vector is 0 = <0, 0, 0>
Given two points
A=(a, b, c) and B=(r, s, t),
then the vector AB in the standard
position is given by
AB = B – A = <r – a, s – b, t – c>.
x
y
z+
2
1
-3
u
-1
-2
1
v
Standard 3D Vectors
b. Find the vector AB if
A=(2, 1, -3) and
B=(-1, -2, 1).
AB = B – A or
AB = <–3, –3, 4>
(2, 1, 0)
AB
Example G:
a. Draw the vectors
u=<2, 1, -3>
v=<-1, -2, 1>

90. Standard 3D Vectors
Scalar-multiplication, vector addition, and the norm
(magnitude) are the same as 2D vectors.

91. Standard 3D Vectors
Scalar-multiplication, vector addition, and the norm
(magnitude) are the same as 2D vectors.
Scalar-multiplication:
Let u=<a, b, c> and λ be a real number, then
λu= λ<a, b, c> = <λa, λb, λc>.

92. Standard 3D Vectors
Scalar-multiplication, vector addition, and the norm
(magnitude) are the same as 2D vectors.
Scalar-multiplication:
Let u=<a, b, c> and λ be a real number, then
λu= λ<a, b, c> = <λa, λb, λc>.
Vector Addition:
Let u = <a, b, c>, v = <r, s, t>, then
u ± v = <a ± r, b ± s, c ± t>

93. Standard 3D Vectors
Scalar-multiplication, vector addition, and the norm
(magnitude) are the same as 2D vectors.
Scalar-multiplication:
Let u=<a, b, c> and λ be a real number, then
λu= λ<a, b, c> = <λa, λb, λc>.
Vector Addition:
Let u = <a, b, c>, v = <r, s, t>, then
u ± v = <a ± r, b ± s, c ± t>
The norm of a vector:
Let u = <a, b, c>, the norm (magnitude or length) of
u is |u| = a2 + b2 + c2

94. In 3D, we set i=<1, 0, 0>, j=<0, 1, 0>, and k=<0, 0, 1>
Standard 3D Vectors

95. In 3D, we set 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
Standard 3D Vectors

96. In 3D, we set 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
Standard 3D Vectors
Example H: Let u=<-1, 1, 2>, v=<1, 3, 3>.
Find and write the linear combination 3u – 2v in terms
of i , j, k, and find |3u – 2v|.

97. In 3D, we set 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
Standard 3D Vectors
Example H: Let u=<-1, 1, 2>, v=<1, 3, 3> and w=<1, 1, 0>
Find and write the linear combination 3u – 2v + w in terms
of i , j, k, and find |3u – 2v + w|.
3u – 2v + w = 3<-1, 1, 2> – 2<1, 3, 3> + <1, 1, 0>
= <-3, 3, 6> + <-2, -6, -6> + <1, 1, 0>
= <-4, -2, 0>
= -4i – 2j + 0k
and that |3u – 2v + w| = 16+4+0 = 20.