Sum of Two Vectors
What is a Vector? A vector is a mathematical object that has both magnitude (length) and direction. We represent vectors as arrows, where the length of the arrow shows the magnitude and the arrowhead shows the direction. Vectors are written as $\vec{a}$, $\vec{b}$, or in component form as $\vec{a} = (a_x, a_y)$.
Definition: Vector Addition
The sum of two vectors $\vec{a}$ and $\vec{b}$ is a new vector $\vec{c} = \vec{a} + \vec{b}$ called the resultant vector. If:
$$ \vec{a} = (a_x, a_y) \quad \text{and} \quad \vec{b} = (b_x, b_y) $$Then their sum is:
$$ \vec{c} = \vec{a} + \vec{b} = (a_x + b_x, \; a_y + b_y) $$We simply add the corresponding components!
Two Geometric Methods:
1. Triangle Rule (Tip-to-Tail)
Place the tail of $\vec{b}$ at the tip of $\vec{a}$. The resultant $\vec{c}$ goes from the tail of $\vec{a}$ to the tip of $\vec{b}$.
2. Parallelogram Rule
Place both vectors at the same starting point. Complete the parallelogram. The diagonal from the common point is the resultant $\vec{c}$.
Interactive: Triangle Rule (Tip-to-Tail Method)
Interactive visualization — available in the full lesson
Drag the colored points to change vectors $\vec{a}$ and $\vec{b}$