Triangle law of vector addition
For 6 and 4 at 60 degrees, R = 8.72, at 23.41° from A.
Vectors
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 6 and vector B of size 4 with 60 degrees between them. The resultant is 8.72, 23.41 degrees from A. The dot product is 12 and the cross product has size 20.78.
Show the formulas
- Resultant
- R = √(A² + B² + 2AB cos θ) = √(6² + 4² + 2 × 6 × 4 × cos 60°) = 8.72
- Direction
- tan β = B sin θ / (A + B cos θ) = 4 × sin 60° / (6 + 4 × cos 60°), so β = 23.41° from A
- Difference
- |A - B| = √(A² + B² - 2AB cos θ) = √(6² + 4² - 2 × 6 × 4 × cos 60°) = 5.29
- Dot product
- A·B = AB cos θ = 6 × 4 × cos 60° = 12
- Cross product
- |A×B| = AB sin θ = 6 × 4 × sin 60° = 20.78
- Components
- A·B = AxBx + AyBy + AzBz = 6 × 2 + 0 × 3.46 + 0 × 0 = 12
- Determinant
- A×B = (AyBz - AzBy) i + (AzBx - AxBz) j + (AxBy - AyBx) k = 20.78k
- Unit vector
- Â = A / |A| = 1i
Step by step
- Given A = 6, B = 4 and the angle between them θ = 60°.
- R² = A² + B² + 2AB cos θ = 6² + 4² + 2 × 6 × 4 × cos 60° = 76
- R = √76 = 8.72
- tan β = B sin θ / (A + B cos θ) = 4 × sin 60° / (6 + 4 × cos 60°) = 0.43
- β = 23.41° from A, towards B.
R for 6 and 4 at other angles
- 0° 100° from A
- 30° 9.6711.93° from A
- 45° 9.2717.76° from A
- 60° 8.7223.41° from A
- 90° 7.2133.69° from A
- 120° 5.2940.89° from A
- 150° 3.2338.26° from A
- 180° 20° from A
Why it works
- Draw A. Start B at the tip of A, keeping its length and direction.
- The side that closes the triangle, from the tail of A to the tip of B, is R = A + B.
- The angle inside the triangle between A and B is 180° - θ, where θ is the angle between them tail to tail.
- The cosine rule gives R² = A² + B² - 2AB cos(180° - θ).
- Since cos(180° - θ) = -cos θ, this is R² = A² + B² + 2AB cos θ.
Triangle law of vector addition: quick answers
What does the triangle law of vector addition say?
If two vectors are drawn head to tail as two sides of a triangle, the third side taken in the opposite order is their resultant, in both size and direction.
How is the angle in the triangle related to θ?
The angle inside the triangle is 180 degrees minus θ, where θ is the angle between the vectors tail to tail. The cosine rule then gives the same R = √(A² + B² + 2AB cos θ).
What is the resultant for 6 and 4 at 60 degrees?
R = 8.72, at 23.41° from A.