1. Vector Addition Visual guide to Vector Addition By Andriy Yamchuk
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License

2. Vectors – Introduction
Vectors describe quantity and direction
Scalars only describe a quantity 𝐴
head
tail
Notation:
Italicized letter with arrow above – 𝑎 or 𝐴
Bold-face letter – 𝒂 or 𝑨
Italicized bold-face letter – 𝒂 or 𝑨
𝐴 (without arrow) – vector magnitude or length
𝜃 – direction with respect to +ive x-axis
Say: “vector A has magnitude of <insert quantity and unit of magnitude A> and directed <insert angle> above x-axis”.
or Say: “vector A has magnitude of <insert quantity and unit of magnitude A> and directed <insert angle> North of East.
Length of a vector is proportional to its magnitude
Drawing a vector:
Position tail at the origin of your co-ordinate system.
Draw vector long enough to represent its magnitude with respect to some scale.
Head should point in the direction described by angle θ.

3. Vector Addition – Geometric Method𝐴 𝐵
𝑅 = 𝐴 + 𝐵
Place vector 𝐴 ’s tail at the origin.
Place vector 𝐵 ’s tail at vector 𝐴 ’s head.
Place resultant vector 𝑅 ’s tail at vector 𝐴 ’s tail and resultant vector 𝑅 ’s head at vector 𝐵 ’s head.

4. Vector Components – Resolving
Complete the right triangle by dropping a perpendicular either on x- or y-axis.
Use trigonometry to find adjacent and opposite sides of this right angle using SOHCAHTOA rule:
Assign appropriate sign to each component based on the quadrant the vector is in. 𝐴
𝐴 𝑥 =𝐴 cos 𝜃
𝐴 𝑦 =𝐴 sin 𝜃
SOH: sin 𝜃 = opposite hypotenuse
CAH: cos 𝜃 = adjacent hypotenuse
TOA: tan 𝜃 = opposite adjacent
Quadrant II 𝐴 𝑥 is -ive
𝐴 𝑦 is +ive
Quadrant I 𝐴 𝑥 is +ive
𝐴 𝑦 is +ive II I
Quadrant III 𝐴 𝑥 is -ive
𝐴 𝑦 is -ive
Quadrant IV 𝐴 𝑥 is +ive
𝐴 𝑦 is -ive
III IV

5. Vector Addition – Component Method
Resolve vectors 𝐴 and 𝐵 into vector components 𝐴 𝑥 , 𝐴 𝑦 , 𝐵 𝑥 and 𝐵 𝑦 .
Add x- and y-components of the two vectors to compute vector components of vector 𝑅 .
𝑅 𝑥 = 𝐴 𝑥 + 𝐵 𝑥
𝑅 𝑦 = 𝐴 𝑦 + 𝐵 𝑦
Remember to include appropriate signs when adding vector components.
Reconstruct vector 𝑅 using trigonometry and Pythagorean theorem:
𝑅 = 𝐴 + 𝐵 𝐴 𝐵
𝐴 𝑥
𝐴 𝑦
𝐵 𝑥
𝐵 𝑦
𝑅 𝑥 𝜃
𝑅 𝑦
𝑅= 𝑅 𝑥 2 + 𝑅 𝑦 2
𝜃= tan −1 𝑅 𝑦 𝑅 𝑥
SOH: sin 𝜃 = opposite hypotenuse
CAH: cos 𝜃 = adjacent hypotenuse
TOA: tan 𝜃 = opposite adjacent