How Do You Add Two Vectors That Point in Different Directions?

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Vectors: Head-to-Tail Addition

Vectors: Head-to-Tail Addition

Lengths in the first row, directions in the second. Every θ is in degrees, counter-clockwise from +x.
|A|m
|B|m
|C|m
|D|m
θA°
0 (+x)
θB°
θC°
θD°
PHYSICS INSIGHTS

Head to tail. Put the tail of each vector on the head of the one before. The resultant R runs from the first tail to the last head: one displacement that does the job of all four. A vector is only a length and a direction, so it can sit anywhere. In Floating Vectors, Shuffle order rebuilds the chain in a new order and R does not move: A + B + C + D = C + A + D + B.

Components. Each θ is measured counter-clockwise from +x, so Vx = |V| cos θ and Vy = |V| sin θ. Add the x-parts, add the y-parts, then |R| = √(Rx² + Ry²). For the direction, find the reference angle tan−1(|Ry| / |Rx|), then place it in the quadrant the signs of Rx and Ry point to.

Subtracting, and closing the loop. A − B = A + (−B): reverse B, then add head to tail as usual. If the last head lands on the first tail, R = 0: zero net displacement, back where you started, and R has no direction. Try |B| = 8.0 m at 180°, with C and D both 3.0 m, at 210° and 30°.

Simulation by The Science Cube — https://www.thesciencecube.com/

Vectors & Vector Addition

What is a Vector?

A vector is any quantity that has both a magnitude (size) and a direction. Displacement, velocity, force, and acceleration are all vectors. A scalar, by contrast, has magnitude only — mass, time, and temperature are scalars.

What is Vector Addition?

When two or more vectors act together, you can replace them with a single equivalent vector called the resultant. The resultant has the same overall effect as all the individual vectors combined.

The Head-to-Tail Method

Place the tail of the second vector at the head (tip) of the first. Repeat for every vector in the chain. The resultant R is the arrow drawn from the tail of the first vector to the head of the last — it is the direct shortcut from start to finish.

How to Use This Simulation

Step-by-Step Mode

  1. Use the sliders to set the magnitude and angle of each vector.
  2. Click Draw A, then Draw B, and so on — build the chain one vector at a time.
  3. Watch Rₓ and Rᵧ update progressively as each vector is added.
  4. Click Draw R at any point after two vectors are drawn to reveal the resultant and the full stepwise calculation.

Floating Vectors Mode

  1. Click Floating Vectors — all four vectors appear scattered randomly on the canvas.
  2. Adjust sliders freely to change any vector while they are floating.
  3. Click Assemble & Draw R — watch the vectors slide smoothly into head-to-tail formation before R is drawn.
  4. This demonstrates a key principle: vector position does not matter — only magnitude and direction determine the resultant.

Keep going:

Equations & Traps: the AP Physics 1 equation deck. See Equations & Traps

AP Physics 1 Mind Maps: every topic on one page. See the Mind Maps

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