lesson
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Imagine walking 4 miles directly east, and then turning to walk 3 miles straight north. How far are you from where you started, and in what overall direction did you travel?
In math and physics, an arrow that has both length and direction is called a vector. The starting dot is its tail (or initial point), and the pointed arrow is its tip (or terminal point).
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So how do we combine two separate journeys into one single combined trip?
The Tip-to-Tail Rule
To add two vectors u and v, place the tail of the second vector directly on the tip of the first vector without changing either vector's direction or length.
The combined sum, called the resultant vector u+v, is the new arrow drawn straight from the starting tail of u to the ending tip of v.
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In the 1840s, Irish mathematician William Rowan Hamilton and later American physicist Josiah Willard Gibbs developed vector geometry to track forces and motion through space with this exact chaining method.
What happens if we add the vectors in reverse order instead?
Order Doesn't Matter
Vector addition is commutative, meaning u+v=v+u.
Whether you walk 4 miles east then 3 miles north, or 3 miles north first and then 4 miles east, you land in the exact same spot.