Numford
Vectors

Resultant of two vectors at 120 degrees

For 10 and 10 at 120 degrees, R = 10, at 60° 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 between120°cos θ = A·B / AB
Unit vectors1iB = -0.5i + 0.87j

Vector A of size 10 and vector B of size 10 with 120 degrees between them. The resultant is 10, 60 degrees from A. The dot product is -50 and the cross product has size 86.6.

Show the formulas
Resultant
R = √(A² + B² + 2AB cos θ) = √(10² + 10² + 2 × 10 × 10 × cos 120°) = 10
Direction
tan β = B sin θ / (A + B cos θ) = 10 × sin 120° / (10 + 10 × cos 120°), so β = 60° from A
Difference
|A - B| = √(A² + B² - 2AB cos θ) = √(10² + 10² - 2 × 10 × 10 × cos 120°) = 17.32
Dot product
A·B = AB cos θ = 10 × 10 × cos 120° = -50
Cross product
|A×B| = AB sin θ = 10 × 10 × sin 120° = 86.6
Components
A·B = AxBx + AyBy + AzBz = 10 × -5 + 0 × 8.66 + 0 × 0 = -50
Determinant
A×B = (AyBz - AzBy) i + (AzBx - AxBz) j + (AxBy - AyBx) k = 86.6k
Unit vector
 = A / |A| = 1i
Working

Step by step

  1. Given A = 10, B = 10 and the angle between them θ = 120°.
  2. R² = A² + B² + 2AB cos θ = 10² + 10² + 2 × 10 × 10 × cos 120° = 100
  3. R = √100 = 10
  4. tan β = B sin θ / (A + B cos θ) = 10 × sin 120° / (10 + 10 × cos 120°) = 1.73
  5. β = 60° from A, towards B.
What the angle does

R for 10 and 10 at other angles

  • 0° 200° from A
  • 30° 19.3215° from A
  • 45° 18.4822.5° from A
  • 60° 17.3230° from A
  • 90° 14.1445° from A
  • 120° 1060° from A
  • 150° 5.1875° from A
  • 180° 0cancels
Derivation

Why it works

  1. Place A and B tail to tail with the angle θ between them, and complete the parallelogram. The diagonal from the common tail is the resultant R.
  2. Drop a perpendicular from the tip of R onto the line of A. B splits into B cos θ along A and B sin θ across it.
  3. The perpendicular triangle gives R² = (A + B cos θ)² + (B sin θ)².
  4. Expand it and use cos²θ + sin²θ = 1 to get R² = A² + B² + 2AB cos θ.
  5. The angle β that R makes with A is the angle of the same triangle: tan β = B sin θ / (A + B cos θ).
Questions

Resultant at 120 degrees: quick answers

What is the resultant of 10 and 10 at 120 degrees?

R = √(10² + 10² + 2 × 10 × 10 × cos 120°) = 10, making 60° with A.

What changes if the angle is not 120 degrees?

Only cos θ changes. The resultant is largest, 20, at 0 degrees and smallest, 0, at 180 degrees. Drag B in the picture to see it.

Which angle does the direction formula give?

tan β = B sin θ / (A + B cos θ) gives the angle between the resultant and the first vector A, measured towards B.

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