AcademyMeasurement and Vectors
Academy
Adding Vectors Geometrically
Level 1 - Physics topic page in Measurement and Vectors.
Principle
Vector addition preserves direction by joining arrows head-to-tail.
Notation
\(\vec{a}\)
first vector
varies
\(\vec{b}\)
second vector
varies
\(\vec{r}\)
resultant vector
same as inputs
\(\theta\)
angle between vectors
rad or deg
Method
The sketch uses the head-to-tail convention: translate the second vector without rotating it, then close the triangle.
The closing arrow has the same start and finish as the two-step path, so it is the resultant.
Resultant
\[\vec{r}=\vec{a}+\vec{b}\]
Triangle size
\[r^2=a^2+b^2+2ab\cos\theta\]
The cosine rule uses the angle between the two vector directions.
Order
\[\vec{a}+\vec{b}=\vec{b}+\vec{a}\]
Changing construction order does not change the final start and finish.
Rules
Vector resultant
\[\vec{r}=\vec{a}+\vec{b}\]
Triangle magnitude
\[r^2=a^2+b^2+2ab\cos\theta\]
Commutative sum
\[\vec{a}+\vec{b}=\vec{b}+\vec{a}\]
Checks
- Do not add magnitudes unless directions match.
- Translating a vector does not change it.
- Resultant units match vector units.
- Reversing a vector changes its sign.