How this instrument works
Combining two arrows means adding their matching components independently: the two x-components add together, and separately, the two y-components add together, producing a single resultant. Unlike ordinary number addition, this is the ONLY operation on this site that takes two full directional quantities as input and produces a third one as output, rather than measuring some single property of one alone.
Geometrically, this matches the tip-to-tail (or parallelogram) rule taught in physics and geometry classes: place the second arrow's tail at the first one's tip, and the sum is the single new arrow drawn straight from the very first tail to the very last tip. The component-by-component method used here always produces exactly that same result, just reached algebraically instead of by drawing.
The origin pair, (0, 0), behaves as this operation's identity element, exactly the way the number 0 behaves in ordinary addition: combining it with anything leaves that quantity completely unchanged. And combining something with its own exact opposite — same magnitude, precisely reversed direction — always cancels to (0, 0), since each pair of matching components sums to zero individually.
- Enter the first vector's components into the Vector 1: x and Vector 1: y fields.
- Enter the second vector's components into the Vector 2: x and Vector 2: y fields.
- Read Sum: x and Sum: y: the sheet adds each pair of matching components independently.
- To combine three or more vectors, add the first two here, then add that resultant to the third vector in a second pass.
Worked example — (1, 2) plus (3, 4)
The pair (1, 2) is combined with (3, 4). The x-components add as 1 + 3 = 4, and the y-components add as 2 + 4 = 6, giving a resultant of (4, 6) — reached by adding each matching pair independently, with no interaction between the x-side and the y-side of the calculation.
Compare combining something with its exact opposite: (−3, 4) plus (3, −4). The x-components cancel, −3 + 3 = 0, and the y-components cancel too, 4 + (−4) = 0, leaving (0, 0) — two equal-length arrows pointing in perfectly opposite directions always sum to nothing at all.
Questions
How do you add two vectors together?
Add each pair of matching components separately: the two x-components sum to the result's x-component, and the two y-components sum to the result's y-component. This component-by-component method is algebraically identical to the geometric tip-to-tail rule.
What is the tip-to-tail rule for combining arrows?
Draw the first arrow, then draw the second one starting from wherever the first one's arrowhead (tip) ended. The sum is the single new arrow drawn directly from the very first tail to the very last tip — a geometric shortcut that always matches the component-by-component method exactly.
What happens when you add the origin pair to something?
Nothing changes — (0, 0) is this operation's identity element, exactly like the number 0 in ordinary arithmetic. Combining it with any quantity returns that same quantity unchanged.
Can two arrows add up to the origin pair?
Yes — whenever the two inputs are exact opposites of each other, meaning they have the same magnitude but point in precisely opposite directions, each pair of matching components sums to zero, and the result is (0, 0).
Is this the same as vector projection?
No — this operation combines two arrows into a new third one by adding their components. Vector projection instead measures how much of ONE arrow points along the direction of another, producing a single scalar length, not a new directional quantity — a fundamentally different operation despite both starting from two inputs.