1. Unit Vectors & Vector Sum
Phys 13 General Physics 1
Unit Vectors & Vector Sum
MARLON FLORES SACEDON

2. What the resultant in 3-Dimensions?
Unit Vector
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?

3. Unit Vector In 3-Dimensions 𝑅 𝑥 =− 𝐹 1𝑥 + 𝐹 2𝑥 +… 𝑅 𝑦 = 𝐹 1𝑦 + 𝐹 2𝑦 +…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 − 𝑅 𝑦 𝑅 𝑥
directions of resultant
𝜑= 𝑇𝑎𝑛 − 𝑅 𝑧 𝑅 𝑥
𝛼= 𝑇𝑎𝑛 − 𝑅 𝑧 𝑅 𝑦

4. Unit Vector 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. Unit Vector Vectors in term of UNIT VECTORS 𝐴 𝐴 + 𝐴 𝑦 𝑗 𝑖 + 𝐴 𝑥
+ 𝐴 𝑦
𝐴 𝑦 𝐴 𝐴
𝐴 𝑦 = 𝐴 𝑦 𝑗 𝑗 𝜃 𝜃 𝑥 𝑖
+ 𝐴 𝑥
𝐴 𝑥
𝐴 𝑥 = 𝐴 𝑥 𝑖
𝐴 = 𝐴 𝑥 + 𝐴 𝑦
𝐴 = 𝐴 𝑥 𝑖 + 𝐴 𝑦 𝑗

6. Unit Vector Vectors in term of UNIT VECTORS In 3-dimensions 𝐴 𝑗 𝑖 𝑘𝑧 𝑥 𝑦
𝐴 = 𝐴 𝑥 + 𝐴 𝑦 + 𝐴 𝑧
𝐴 𝑧 𝑘
𝐴 𝑦 𝑗
𝐴 = 𝐴 𝑥 𝑖 + 𝐴 𝑦 𝑗 + 𝐴 𝑧 𝑘 𝐴 𝑗
𝐴 𝑥 𝑖 𝑘 𝑖

7. Vector sum Vector Sum in terms of unit vector Suppose: 𝐴 + 𝐵 = 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 sum Vector Sum in terms of unit vector Example: (a) 𝐶 + 𝐷 =
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. eNd