1. Phys 13 General Physics 1
Vector Product
MARLON FLORES SACEDON

2. Vector product 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

3. Vector product 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

4. 𝐴 ∙ 𝐵 =𝐴𝐵𝑐𝑜𝑠𝜃 Vector product
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

5. Vector product 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

6. Vector product Product of vectors: The scalar product or Dot product 𝐻𝑧 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

7. 𝐴 𝑥 𝐵 = 𝐴 𝐵 𝑠𝑖𝑛𝜃 𝑛 Vector product
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

8. 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

9. Vector product 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

10. Vector product 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

11. Vector product 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

12. Vector product 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

13. Assignment 𝐵 𝐴 𝐶 1) Given: 2) Figure: 300𝑔𝑓 Find: Find: 100𝑔𝑓 160𝑔𝑓
𝐿 = 𝑖 +6 𝑗 −3 𝑘 𝐵 𝐴
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 𝐶
Find:
e) 𝐿 × 𝑀 × 𝑁
a) Convert the three vectors in terms of unit vectors.
f) 5 𝑀 × 𝑁
b) 𝐴 + 𝐵 + 𝐶 Note: Use the converted vectors in terms of unit vectors.
g) 5 𝑁 × 𝑀

14. eNd