Numford
Vectors

Polygon law of vector addition

Three vectors added head to tail give R = 5.21 at 43.68° to the x axis.

ABCR
A, B and C head to tail. R closes the polygon.

Try two of them

Drag the glowing tips, or type values. Angles are measured from the x axis, and θ is the angle from A to B. Pick a result card to see it built in the picture.

Angle between60°cos θ = A·B / AB
Unit vectors1iB = 0.5i + 0.87j

Vector A of size 4 and vector B of size 3 with 60 degrees between them. The resultant is 6.08, 25.28 degrees from A. The dot product is 6 and the cross product has size 10.39.

Show the formulas
Resultant
R = √(A² + B² + 2AB cos θ) = √(4² + 3² + 2 × 4 × 3 × cos 60°) = 6.08
Direction
tan β = B sin θ / (A + B cos θ) = 3 × sin 60° / (4 + 3 × cos 60°), so β = 25.28° from A
Difference
|A - B| = √(A² + B² - 2AB cos θ) = √(4² + 3² - 2 × 4 × 3 × cos 60°) = 3.61
Dot product
A·B = AB cos θ = 4 × 3 × cos 60° = 6
Cross product
|A×B| = AB sin θ = 4 × 3 × sin 60° = 10.39
Components
A·B = AxBx + AyBy + AzBz = 4 × 1.5 + 0 × 2.6 + 0 × 0 = 6
Determinant
A×B = (AyBz - AzBy) i + (AzBx - AxBz) j + (AxBy - AyBx) k = 10.39k
Unit vector
 = A / |A| = 1i
Working

Step by step

  1. Given A = 4i, B = 1.5i + 2.6j, C = -1.73i + 1j.
  2. Rx = 4 + 1.5 + -1.73 = 3.77
  3. Ry = 0 + 2.6 + 1 = 3.6
  4. R = √(14.2 + 12.95) = 5.21
  5. tan α = Ry / Rx = 0.95, so R makes 43.68° with the x axis.
Derivation

Why it works

  1. Draw the first vector. Start the second at its tip, the third at the tip of the second, and so on.
  2. The vector from the tail of the first to the head of the last is the resultant R.
  3. Add the components: Rx = Ax + Bx + Cx and Ry = Ay + By + Cy.
  4. The magnitude is R = √(Rx² + Ry²) and the direction is tan α = Ry / Rx.
  5. The order of addition does not change the result, so the polygon can be drawn in any order.
Questions

Polygon law of vectors: quick answers

What does the polygon law of vector addition say?

If several vectors are drawn head to tail, the vector from the tail of the first to the head of the last is their resultant.

What is the resultant of the three vectors shown?

Rx = 3.77 and Ry = 3.6, so R = 5.21.

Does the order of the vectors matter?

No. Vector addition is commutative and associative, so any order gives the same closing side.

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