Resultant of two vectors
For 8 and 6 at 60 degrees, R = 12.17, at 25.28° 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 8 and vector B of size 6 with 60 degrees between them. The resultant is 12.17, 25.28 degrees from A. The dot product is 24 and the cross product has size 41.57.
Show the formulas
- Resultant
- R = √(A² + B² + 2AB cos θ) = √(8² + 6² + 2 × 8 × 6 × cos 60°) = 12.17
- Direction
- tan β = B sin θ / (A + B cos θ) = 6 × sin 60° / (8 + 6 × cos 60°), so β = 25.28° from A
- Difference
- |A - B| = √(A² + B² - 2AB cos θ) = √(8² + 6² - 2 × 8 × 6 × cos 60°) = 7.21
- Dot product
- A·B = AB cos θ = 8 × 6 × cos 60° = 24
- Cross product
- |A×B| = AB sin θ = 8 × 6 × sin 60° = 41.57
- Components
- A·B = AxBx + AyBy + AzBz = 8 × 3 + 0 × 5.2 + 0 × 0 = 24
- Determinant
- A×B = (AyBz - AzBy) i + (AzBx - AxBz) j + (AxBy - AyBx) k = 41.57k
- Unit vector
- Â = A / |A| = 1i
Step by step
- Given A = 8, B = 6 and the angle between them θ = 60°.
- R² = A² + B² + 2AB cos θ = 8² + 6² + 2 × 8 × 6 × cos 60° = 148
- R = √148 = 12.17
- tan β = B sin θ / (A + B cos θ) = 6 × sin 60° / (8 + 6 × cos 60°) = 0.47
- β = 25.28° from A, towards B.
R for 8 and 6 at other angles
- 0° 140° from A
- 30° 13.5312.81° from A
- 45° 12.9619.11° from A
- 60° 12.1725.28° from A
- 90° 1036.87° from A
- 120° 7.2146.1° from A
- 150° 4.1146.94° from A
- 180° 20° from A
Why it works
- 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.
- 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.
- The perpendicular triangle gives R² = (A + B cos θ)² + (B sin θ)².
- Expand it and use cos²θ + sin²θ = 1 to get R² = A² + B² + 2AB cos θ.
- The angle β that R makes with A is the angle of the same triangle: tan β = B sin θ / (A + B cos θ).
Resultant of two vectors: quick answers
What is the resultant of 8 and 6 at 60 degrees?
R = √(8² + 6² + 2 × 8 × 6 × cos 60°) = 12.17, making 25.28° with A.
What changes if the angle is not 60 degrees?
Only cos θ changes. The resultant is largest, 14, at 0 degrees and smallest, 2, 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.