Sum of Several Vectors
Big Idea:
Vectors are mathematical objects that have both magnitude (length) and direction. When we add several vectors together, we combine their effects to find a single resultant vector that represents the total combined effect.
📐 Formal Definition
Given vectors $\vec{a}_1, \vec{a}_2, \vec{a}_3, \ldots, \vec{a}_n$, their sum (or resultant) is defined as:
$$\vec{S} = \vec{a}_1 + \vec{a}_2 + \vec{a}_3 + \cdots + \vec{a}_n = \sum_{i=1}^{n} \vec{a}_i$$
The resultant vector $\vec{S}$ connects the starting point of the first vector to the endpoint of the last vector when they are placed head-to-tail.
📏 Key Properties
- Commutative: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$
Order doesn't matter! - Associative: $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$
Grouping doesn't matter! - Zero Vector: $\vec{a} + \vec{0} = \vec{a}$
- Inverse: $\vec{a} + (-\vec{a}) = \vec{0}$
- Closed Polygon Rule: If vectors form a closed polygon, their sum is $\vec{0}$.