How do I add two vectors by components and check the result geometrically?
Quick answer
To add two 2D vectors, add their horizontal components and their vertical components separately: if A = (Ax, Ay) and B = (Bx, By), then A + B = (Ax + Bx, Ay + By). For example, (2, 1) + (3, 4) = (5, 5). Geometrically, place B’s tail at A’s tip; the sum runs from A’s starting point to B’s final tip. [S1][S2]
The question, the details, the next step.
I have two vectors written as pairs of numbers. What does adding them by components mean?
Treat each pair as a horizontal and vertical change. Add the horizontal entries together, then add the vertical entries:
- If A = (Ax, Ay) and B = (Bx, By), then A + B = (Ax + Bx, Ay + By).
- Keep the positions aligned: first with first, second with second.
For instance, take A = (2, 1) and B = (3, 4). The horizontal total is 2 + 3 = 5, and the vertical total is 1 + 4 = 5, so A + B = (5, 5). This is componentwise vector addition. [S1]
The components represent changes along the two axes. Keep each component in its original position when you write the result.
How can I check that (5, 5) makes sense without relying only on the arithmetic?
Draw a coordinate grid. Start vector A at the origin and draw it to (2, 1). Then move vector B without changing its direction or length: put B’s tail at A’s tip, so it goes from (2, 1) to (5, 5). The overall vector starts at the origin and ends at (5, 5). That start-to-finish arrow is the resultant, the geometric sum. [S2]
Check the endpoint in both directions: B adds 3 horizontally and 4 vertically to A’s endpoint. If your sketch finishes at (5, 5), it agrees with the component calculation. The tip-to-tail construction connects the initial position to the final position. [S1][S2]
What if one of my components is negative, or I draw the vectors in a different order?
Keep a negative sign as part of the component when you add. For example, if the horizontal components are 2 and −3, their horizontal sum is 2 + (−3) = −1. Pair that result with the sum of the vertical components; do not switch the component positions.
For a clear check, draw the vectors in the same order as your calculation: put the second vector’s tail at the first vector’s tip. The overall arrow runs from the original starting point to the final endpoint. Compare that endpoint with the ordered pair you calculated. [S1][S2]
Can you give me a reusable way to show my work so I can spot an error?
Use this copyable layout, replacing the letters and values with your own:
A = (Ax, Ay) = (horizontal value, vertical value) B = (Bx, By) = (horizontal value, vertical value) A + B = (Ax + Bx, Ay + By) = (horizontal sum, vertical sum)
Geometric check: Draw A from your starting point. Place B’s tail at A’s tip. The resultant should run from A’s starting point to the final tip, matching the calculated pair. [S1][S2]
For the worked example, write A = (2, 1), B = (3, 4), and A + B = (2 + 3, 1 + 4) = (5, 5). In the sketch, A ends at (2, 1); B then ends at (5, 5); the resultant goes from (0, 0) to (5, 5). If the endpoint differs, recheck the component pairing, arithmetic, and signs.
A practical checklist
Write each vector as an ordered pair and identify which entry is horizontal and which is vertical.
Add horizontal components together and vertical components together, preserving signs.
Write the result as an ordered pair in the same component order.
Sketch the first vector, then place the second vector’s tail at the first vector’s tip.
Check that the resultant runs from the original start to the final tip and matches the calculated endpoint.
A few more useful answers
Do I add the first component of one vector to the second component of the other?
No. Add matching positions: horizontal to horizontal and vertical to vertical. Mixing component positions changes the componentwise calculation. [S1]
Does a rough drawing have to be perfectly to scale?
What does the resultant represent in the tip-to-tail drawing?
It is the single vector from the first vector’s starting point to the final tip after the second vector is placed head to tail. [S2]
Sources
References used for this guide.
- [S1] OpenStax University Physics Volume 1: Algebra of Vectors openstax.org · Checked
- [S2] OpenStax University Physics Volume 1: Scalars and Vectors openstax.org · Checked