Numford
Vectors

Resultant of two vectors at 60 degrees

For 10 and 10 at 60 degrees, R = 17.32, at 30° 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 10 and vector B of size 10 with 60 degrees between them. The resultant is 17.32, 30 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 60°) = 17.32
Direction
tan β = B sin θ / (A + B cos θ) = 10 × sin 60° / (10 + 10 × cos 60°), so β = 30° from A
Difference
|A - B| = √(A² + B² - 2AB cos θ) = √(10² + 10² - 2 × 10 × 10 × cos 60°) = 10
Dot product
A·B = AB cos θ = 10 × 10 × cos 60° = 50
Cross product
|A×B| = AB sin θ = 10 × 10 × sin 60° = 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 = 60°.
  2. R² = A² + B² + 2AB cos θ = 10² + 10² + 2 × 10 × 10 × cos 60° = 300
  3. R = √300 = 17.32
  4. tan β = B sin θ / (A + B cos θ) = 10 × sin 60° / (10 + 10 × cos 60°) = 0.58
  5. β = 30° 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 60 degrees: quick answers

What is the resultant of 10 and 10 at 60 degrees?

R = √(10² + 10² + 2 × 10 × 10 × cos 60°) = 17.32, making 30° with A.

What changes if the angle is not 60 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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