1. Physics 133 Electromagnetism
Vectors Analysis
MARLON FLORES SACEDON

2. What the resultant in 3-Dimensions?
Vector analysis
Recall the component method:
𝑅 𝑥 =− 𝐹 1𝑥 + 𝐹 2𝑥 +…
𝐹 1 𝜃 𝛽
𝐹 2 𝑦 𝑥
𝑅 𝑦 = 𝐹 1𝑦 + 𝐹 2𝑦 +…
𝑅= Σ𝑅 𝑥 Σ𝑅 𝑦 2
Magnitude of resultant
𝛾= 𝑇𝑎𝑛 −1 𝑅 𝑦 𝑅 𝑥
direction of resultant
These formulas are for 2-Dimensions only!
What the resultant in 3-Dimensions? 𝑧

3. Vector analysis In 3-Dimensions 𝑅 𝑥 =− 𝐹 1𝑥 + 𝐹 2𝑥 +…𝑧 𝑥 𝑦
𝑅 𝑥 =− 𝐹 1𝑥 + 𝐹 2𝑥 +…
𝑅 𝑦 = 𝐹 1𝑦 + 𝐹 2𝑦 +…
𝑅 𝑧 = 𝐹 1𝑧 + 𝐹 2𝑧 +…
𝐹 1
Magnitude of resultant
𝑅= Σ𝑅 𝑥 Σ𝑅 𝑦 Σ𝑅 𝑧 2
𝐹 2
𝛾= 𝑇𝑎𝑛 −1 𝐴 𝑦 𝐴 𝑥
directions of resultant
𝜑= 𝑇𝑎𝑛 −1 𝐴 𝑧 𝐴 𝑥
𝛼= 𝑇𝑎𝑛 −1 𝐴 𝑧 𝐴 𝑦

4. Vector analysis Vectors in term of UNIT VECTORS In 3-dimensions
What is Unit Vector? 𝑧 𝑥 𝑦
A unit vector is a vector that has a magnitude of 1, with no units. Its only purpose is to point—that is, to describe a direction in space. A unit vector is often denoted by a lowercase letter with a circumflex, or caret, or "hat": (∧). 𝑗 𝑢
: Unit vector of vector 𝑢
𝑢 : Vector 𝑢 𝑘 𝑖
𝑢 : Magnitude of vector 𝑢
Normalized vectors
𝑢 = 𝑢 𝑢

5. Vector analysis Vectors in term of UNIT VECTORS 𝑟 𝑟 + 𝑟 𝑦 𝑗 𝑖 + 𝑟 𝑥
+ 𝑟 𝑦
𝑟 𝑦 𝑟 𝑟
𝑟 𝑦 = 𝑟 𝑦 𝑗 𝑗 𝜃 𝜃 𝑥 𝑖
+ 𝑟 𝑥
𝑟 𝑥
𝑟 𝑥 = 𝑟 𝑥 𝑖
𝑟 = 𝑟 𝑥 + 𝑟 𝑦
𝑟 = 𝑟 𝑥 𝑖 + 𝑟 𝑦 𝑗

6. Vector analysis Vectors in term of UNIT VECTORS In 3-dimensions 𝑟 𝑗 𝑖𝑧 𝑥 𝑦
𝑟 = 𝑟 𝑥 + 𝑟 𝑦 + 𝑟 𝑧
𝑟 𝑧 𝑘
𝑟 𝑦 𝑗
𝑟 = 𝑟 𝑥 𝑖 + 𝑟 𝑦 𝑗 + 𝑟 𝑧 𝑘 𝑟 𝑗
𝑟 𝑥 𝑖 𝑘 𝑖

7. Vector analysis Vector Sum in terms of unit vector Suppose: 𝐴 + 𝐵 =
Find:
Addition of vectors: 𝐴 + 𝐵
Subtraction of vectors: 𝐴 − 𝐵
𝐴 = 𝐴 𝑥 𝑖 + 𝐴 𝑦 𝑗 + 𝐴 𝑧 𝑘
𝐵 = 𝐵 𝑥 𝑖 + 𝐵 𝑦 𝑗 + 𝐵 𝑧 𝑘
Addition of vectors: 𝐴 + 𝐵
𝐴 + 𝐵 =
𝐴 𝑥 𝑖 + 𝐴 𝑦 𝑗 + 𝐴 𝑧 𝑘
+ 𝐵 𝑥 𝑖 + 𝐵 𝑦 𝑗 + 𝐵 𝑧 𝑘
= 𝐴 𝑥 + 𝐵 𝑥 𝑖 + 𝐴 𝑦 + 𝐵 𝑦 𝑗 + 𝐴 𝑧 + 𝐵 𝑧 𝑘
(b) Subtraction of vectors: 𝐴 − 𝐵
𝐴 − 𝐵 =
𝐴 𝑥 𝑖 + 𝐴 𝑦 𝑗 + 𝐴 𝑧 𝑘
− 𝐵 𝑥 𝑖 + 𝐵 𝑦 𝑗 + 𝐵 𝑧 𝑘
= 𝐴 𝑥 − 𝐵 𝑥 𝑖 + 𝐴 𝑦 − 𝐵 𝑦 𝑗 + 𝐴 𝑧 − 𝐵 𝑧 𝑘

8. Vector analysis Vector Sum in terms of unit vector Example:
5 𝑖 −3 𝑗 + −6 𝑖 + 𝑗 − 𝑘
Example:
= 5−6 𝑖 + −3+1 𝑗 + 0− 𝑘
𝐶 =5 𝑖 −3 𝑗
𝐷 =−6 𝑖 + 𝑗 − 𝑘
=− 𝑖 −2 𝑗 − 𝑘
=− 𝑖 +2 𝑗 𝑘
ANSWER
Find:
𝐶 + 𝐷
(b) 𝐶 − 𝐷
Normalized vectors 𝐶 & 𝐷
(b) 𝐶 − 𝐷 =
5 𝑖 −3 𝑗 − −6 𝑖 + 𝑗 − 𝑘
=5 𝑖 −3 𝑗 +6 𝑖 − 𝑗 𝑘
=11 𝑖 −4 𝑗 𝑘
Note: Normalizing is the same as finding the
equivalent unit vector of a given vector.
ANSWER
(c) 𝐶 = 𝐶 𝐶 = 5 𝑖 −3 𝑗 − =0.86 𝑖 −0.51 𝑗
𝐷 = 𝐷 𝐷 = −6 𝑖 + 𝑗 − 𝑘 − − =−0.98 𝑖 𝑗 −0.08 𝑘 =− 0.98 𝑖 −0.16 𝑗 𝑘

9. Vector analysis Product of vectors Scalar product or Dot product
Vector product or Cross product
Ex. 𝐴 dot 𝐵 is equal to 𝐴 ∙ 𝐵
Ex. 𝐴 cross 𝐵 is equal to 𝐴 × 𝐵

10. Vector analysis Product of vectors: The scalar product or Dot product
𝐴 ∙ 𝐵 to be the magnitude of 𝐴 multiplied by the component of 𝐵 in the direction of 𝐴 .
𝐵 ∙ 𝐴 to be the magnitude of 𝐵 multiplied by the component of 𝐴 in the direction of 𝐵 .
𝐵𝑐𝑜𝑠𝜃 𝐴 𝐵 𝜃 𝜃 𝐴 𝐵
𝐴𝑐𝑜𝑠𝜃
𝐵 ∙ 𝐴 =𝐵𝐴𝑐𝑜𝑠𝜃
𝐴 ∙ 𝐵 =𝐴𝐵𝑐𝑜𝑠𝜃
𝐴 ∙ 𝐵 = 𝐴 𝐵 𝑐𝑜𝑠𝜃
𝐵 ∙ 𝐴 = 𝐵 𝐴 𝑐𝑜𝑠𝜃
So, therefore:
𝐴 ∙ 𝐵 = 𝐵 ∙ 𝐴

11. 𝐴 ∙ 𝐵 =𝐴𝐵𝑐𝑜𝑠𝜃 Vector analysis
Product of vectors: The scalar product or Dot product
So, the definition of Scalar or Dot product is,
𝐴 ∙ 𝐵 =𝐴𝐵𝑐𝑜𝑠𝜃
The scalar product of unit vectors
𝑖 ∙ 𝑖
= 𝑖 𝑖 𝑐𝑜𝑠𝜃
=(1)(1) 𝑐𝑜𝑠 0 𝑜 =1
𝑗 ∙ 𝑗 =1
𝑘 ∙ 𝑘 =1
𝑖 ∙ 𝑗 =0
𝑗 ∙ 𝑖 =0
𝑗 ∙ 𝑘 =0
𝑖 ∙ 𝑘 =0
𝑘 ∙ 𝑖 =0
𝑘 ∙ 𝑗 =0

12. Vector analysis Product of vectors: The scalar product or Dot product
Suppose:
Find:
𝐴 ∙ 𝐵
𝐴 = 𝐴 𝑥 𝑖 + 𝐴 𝑦 𝑗 + 𝐴 𝑧 𝑘
𝐵 = 𝐵 𝑥 𝑖 + 𝐵 𝑦 𝑗 + 𝐵 𝑧 𝑘
Then:
𝐴 ∙ 𝐵 =
𝐴 𝑥 𝑖 + 𝐴 𝑦 𝑗 + 𝐴 𝑧 𝑘
∙ 𝐵 𝑥 𝑖 + 𝐵 𝑦 𝑗 + 𝐵 𝑧 𝑘 1 1
= 𝐴 𝑥 𝐵 𝑥 𝑖 ∙ 𝑖
+ 𝐴 𝑥 𝐵 𝑦 𝑖 ∙ 𝑗
+ 𝐴 𝑥 𝐵 𝑧 𝑖 ∙ 𝑘
+ 𝐴 𝑦 𝐵 𝑥 𝑗 ∙ 𝑖
+ 𝐴 𝑦 𝐵 𝑦 𝑗 ∙ 𝑗
+ 𝐴 𝑦 𝐵 𝑧 𝑗 ∙ 𝑘 1
+ 𝐴 𝑧 𝐵 𝑥 𝑘 ∙ 𝑖
+ 𝐴 𝑧 𝐵 𝑦 𝑘 ∙ 𝑗
+ 𝐴 𝑧 𝐵 𝑧 𝑘 ∙ 𝑘
𝐴 ∙ 𝐵 = 𝐴 𝑥 𝐵 𝑥 + 𝐴 𝑦 𝐵 𝑦 + 𝐴 𝑧 𝐵 𝑧
Therefore:
So:
ABcosθ= 𝐴 𝑥 𝐵 𝑥 + 𝐴 𝑦 𝐵 𝑦 + 𝐴 𝑧 𝐵 𝑧
𝜃= 𝑐𝑜𝑠 −1 𝐴 𝑥 𝐵 𝑥 + 𝐴 𝑦 𝐵 𝑦 + 𝐴 𝑧 𝐵 𝑧 𝐴𝐵
Angle between vectors

13. Vector analysis Product of vectors: The scalar product or Dot product𝑧 𝑥 𝑦
Example:
Find:
Given:
𝐺 =9 𝑖 −3 𝑗 + 𝑘
a) 𝐺 ∙ 𝐻 𝐻
𝐻 = 𝑖 −6 𝑘
b) Angle between vectors 𝐺 and 𝐻
−6 𝑘 𝜃
Solution: 𝑖
9 𝑖 𝐺
−3 𝑗
a) 𝐺 ∙ 𝐻 =
𝐺 𝑥 𝐻 𝑥 + 𝐺 𝑦 𝐻 𝑦 + 𝐺 𝑧 𝐻 𝑧 𝑘
= 9 1
+ −3 0
+(1)(−6) =3
ANSWER
𝑐𝑜𝑠 −1 𝐺 𝑥 𝐻 𝑥 + 𝐺 𝑦 𝐻 𝑦 + 𝐺 𝑧 𝐻 𝑧 𝐺𝐻
b) 𝜃=
=𝑐𝑜𝑠 − − −6 2
=𝑐𝑜𝑠 −1 [ ]
𝜃= 𝑜
ANSWER

14. 𝐴 𝑥 𝐵 = 𝐴 𝐵 𝑠𝑖𝑛𝜃 𝑛 Vector analysis
Product of vectors: The vector product or Cross product 𝑛 𝑛 𝐵 𝐴 𝜃 𝐵 𝐴 𝜃
𝐵 𝑠𝑖𝑛𝜃
𝐵 𝑥 𝐴
What is 𝐴 cross 𝐵 is equal to 𝐴 × 𝐵 ?
ANSWER: 𝐴 × 𝐵 to be the magnitude of 𝐴 multiplied by the component of 𝐵 perpendicular to 𝐴 , then multiplied by a unit vector 𝑛 which is normal to swipe area of the two vectors.
𝐵 𝑥 𝐴 =− 𝐴 𝑥 𝐵
The property of vector product.
𝐴 𝑥 𝐵 = 𝐴 𝐵 𝑠𝑖𝑛𝜃 𝑛

15. Vector analysis
Product of vectors: The vector product or Cross product
The vector product of unit vectors
𝑖 𝑥 𝑖 =1 1 𝑠𝑖𝑛0 𝑛 =0
𝑖 𝑥 𝑗 =1 1 𝑠𝑖𝑛90 𝑘 = 𝑘
𝑗 𝑥 𝑖 =1 1 𝑠𝑖𝑛90 𝑘 =− 𝑘
𝑘 𝑥 𝑘 =0
𝑖 𝑥 𝑖 =0
𝑗 𝑥 𝑗 =0
𝑖 𝑥 𝑗 = 𝑘
𝑗 𝑥 𝑖 =− 𝑘
𝑘 𝑥 𝑗 =− 𝑖
𝑖 𝑥 𝑘 =− 𝑗
𝑗 𝑥 𝑘 = 𝑖
𝑘 𝑥 𝑖 = 𝑗

16. Vector analysis
Product of vectors: The vector product or Cross product
Suppose:
Find:
𝐴 𝑥 𝐵
𝐴 = 𝐴 𝑥 𝑖 + 𝐴 𝑦 𝑗 + 𝐴 𝑧 𝑘
𝐵 = 𝐵 𝑥 𝑖 + 𝐵 𝑦 𝑗 + 𝐵 𝑧 𝑘
Evaluate:
𝐴 𝑥 𝐵 =
𝐴 𝑥 𝑖 + 𝐴 𝑦 𝑗 + 𝐴 𝑧 𝑘
𝑥 𝐵 𝑥 𝑖 + 𝐵 𝑦 𝑗 + 𝐵 𝑧 𝑘 𝑘
− 𝑗
= 𝐴 𝑥 𝐵 𝑥 𝑖 𝑥 𝑖
+ 𝐴 𝑥 𝐵 𝑦 𝑖 𝑥 𝑗
+ 𝐴 𝑥 𝐵 𝑧 𝑖 𝑥 𝑘
− 𝑘 𝑖
+ 𝐴 𝑦 𝐵 𝑥 𝑗 𝑥 𝑖
+ 𝐴 𝑦 𝐵 𝑦 𝑗 𝑥 𝑗
+ 𝐴 𝑦 𝐵 𝑧 𝑗 𝑥 𝑘
+ 𝐴 𝑧 𝐵 𝑥 𝑘 𝑥 𝑖 𝑗
+ 𝐴 𝑧 𝐵 𝑦 𝑘 𝑥 𝑗
− 𝑖
+ 𝐴 𝑧 𝐵 𝑧 𝑘 𝑥 𝑘
= 𝐴 𝑥 𝐵 𝑦 𝑘 + 𝐴 𝑥 𝐵 𝑧 − 𝑗 + 𝐴 𝑦 𝐵 𝑥 − 𝑘 + 𝐴 𝑦 𝐵 𝑧 𝑖 + 𝐴 𝑧 𝐵 𝑥 𝑗 + 𝐴 𝑧 𝐵 𝑦 − 𝑖
𝐴 𝑥 𝐵 = 𝐴 𝑦 𝐵 𝑧 𝑖 + 𝐴 𝑧 𝐵 𝑥 𝑗 + 𝐴 𝑥 𝐵 𝑦 𝑘 − 𝐴 𝑦 𝐵 𝑥 𝑘 − 𝐴 𝑧 𝐵 𝑦 𝑖 − 𝐴 𝑥 𝐵 𝑧 𝑗
𝐴 𝑥 𝐵 = 𝑖 𝑗 𝑘 𝐴 𝑥 𝐴 𝑦 𝐴 𝑧 𝐵 𝑥 𝐵 𝑦 𝐵 𝑧
𝐴 𝑥 𝐵 = 𝑖 𝑗 𝑘 𝐴 𝑥 𝐴 𝑦 𝐴 𝑧 𝐵 𝑥 𝐵 𝑦 𝐵 𝑧
𝑖 𝑗 𝐴 𝑥 𝐴 𝑦 𝐵 𝑥 𝐵 𝑦

17. Vector analysis
Product of vectors: The vector product or Cross product
𝐴 𝑥 𝐵 = 𝐴 𝑦 𝐵 𝑧 𝑖 + 𝐴 𝑧 𝐵 𝑥 𝑗 + 𝐴 𝑥 𝐵 𝑦 𝑘 − 𝐴 𝑦 𝐵 𝑥 𝑘 − 𝐴 𝑧 𝐵 𝑦 𝑖 − 𝐴 𝑥 𝐵 𝑧 𝑗
𝐴 𝑥 𝐵 = 𝑖 𝑗 𝑘 𝐴 𝑥 𝐴 𝑦 𝐴 𝑧 𝐵 𝑥 𝐵 𝑦 𝐵 𝑧

18. Vector analysis
Product of vectors: The vector product or Cross product
𝐴 𝑥 𝐵 = 𝐴 𝑦 𝐵 𝑧 𝑖 + 𝐴 𝑧 𝐵 𝑥 𝑗 + 𝐴 𝑥 𝐵 𝑦 𝑘 − 𝐴 𝑦 𝐵 𝑥 𝑘 − 𝐴 𝑧 𝐵 𝑦 𝑖 − 𝐴 𝑥 𝐵 𝑧 𝑗
𝐴 𝑥 𝐵 = 𝑖 𝑗 𝑘 𝐴 𝑥 𝐴 𝑦 𝐴 𝑧 𝐵 𝑥 𝐵 𝑦 𝐵 𝑧
𝐴 𝑥 𝐵 = 𝐴 𝑦 𝐵 𝑧 𝑖 − 𝐴 𝑧 𝐵 𝑦 𝑖 + 𝐴 𝑧 𝐵 𝑥 𝑗 − 𝐴 𝑥 𝐵 𝑧 𝑗 + 𝐴 𝑥 𝐵 𝑦 𝑘 − 𝐴 𝑦 𝐵 𝑥 𝑘
𝐴 𝑥 𝐵 = 𝐴 𝑦 𝐵 𝑧 − 𝐴 𝑧 𝐵 𝑦 𝑖 + 𝐴 𝑧 𝐵 𝑥 − 𝐴 𝑥 𝐵 𝑧 𝑗 + 𝐴 𝑥 𝐵 𝑦 − 𝐴 𝑦 𝐵 𝑥 𝑘
𝐴 𝑥 𝐵 = 𝑖 𝑗 𝑘 𝐴 𝑥 𝐴 𝑦 𝐴 𝑧 𝐵 𝑥 𝐵 𝑦 𝐵 𝑧

19. Vector analysis
Product of vectors: The vector product or Cross product
Example:
Given:
Find:
𝐹 =3 𝑖 −5 𝑗 +2 𝑘
a) 𝐹 × 𝐺
b) 𝐺 × 𝐹
𝐺 = 𝑗 −7 𝑘
𝐹 𝑥 𝐺 = 𝑖 𝑗 𝑘 3 − −7
= 35−2 𝑖 𝑗
+ 3−0 𝑘
=33 𝑖 +21 𝑗 +3 𝑘
𝐺 𝑥 𝐹 = 𝑖 𝑗 𝑘 0 1 −7 3 −5 2
= 2−35 𝑖
+ −21−0 𝑗
+ 0−3 𝑘
=−33 𝑖 −21 𝑗 −3 𝑘
=− 33 𝑖 +21 𝑗 +3 𝑘
So:
𝐺 × 𝐹 =− 𝐹 × 𝐺

20. Assignment 𝐴 𝐵 𝐶 1) Given: 2) Figure: Find: 300𝑔𝑓 Find: 100𝑔𝑓 160𝑔𝑓
𝐿 = 𝑖 +6 𝑗 −3 𝑘 𝐴 𝐵
𝑀 =6 𝑖 − 𝑗 +7 𝑘
100𝑔𝑓
25 𝑜
160𝑔𝑓
𝑁 =−7 𝑘
65 𝑜
300𝑔𝑓
40 𝑜
Find:
a) 𝐿 × 𝑁
b) 𝐿 × 𝑀
c) 𝐿 ∙ 𝑀 × 𝑁
d) 𝐿 × 𝑀 ∙ 𝑀 × 𝑁
e) 𝐿 × 𝑀 × 𝑁 𝐶
Find:
f) 5 𝑀 × 𝑁
a) Convert the three vectors in terms of unit vectors.
g) 5 𝑁 × 𝑀
b) Find the vector sum ( 𝐴 + 𝐵 + 𝐶 ) Note: Use the converted vectors in terms of unit vectors to find the sum.
h) Find the unit vectors of 𝐿 , 𝑁 and 𝑀 .
i) Graph the vectors 𝐿 , 𝑁 and 𝑀 .

21. eNd