Vectors

Resultant of two equal vectors

For 6 and 6 at 70 degrees, R = 9.83, at 35° 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 between70°cos θ = A·B / AB

Unit vectors1iB = 0.34i + 0.94j

Vector A of size 6 and vector B of size 6 with 70 degrees between them. The resultant is 9.83, 35 degrees from A. The dot product is 12.31 and the cross product has size 33.83.

Show the formulas

Resultant: R = √(A² + B² + 2AB cos θ) = √(6² + 6² + 2 × 6 × 6 × cos 70°) = 9.83
Direction: tan β = B sin θ / (A + B cos θ) = 6 × sin 70° / (6 + 6 × cos 70°), so β = 35° from A
Difference: |A - B| = √(A² + B² - 2AB cos θ) = √(6² + 6² - 2 × 6 × 6 × cos 70°) = 6.88
Dot product: A·B = AB cos θ = 6 × 6 × cos 70° = 12.31
Cross product: |A×B| = AB sin θ = 6 × 6 × sin 70° = 33.83
Components: A·B = AxBx + AyBy + AzBz = 6 × 2.05 + 0 × 5.64 + 0 × 0 = 12.31
Determinant: A×B = (AyBz - AzBy) i + (AzBx - AxBz) j + (AxBy - AyBx) k = 33.83k
Unit vector: Â = A / |A| = 1i

Working

Step by step

Given A = 6, B = 6 and the angle between them θ = 70°.
R² = A² + B² + 2AB cos θ = 6² + 6² + 2 × 6 × 6 × cos 70° = 96.63
For equal vectors this is R = 2A cos(θ/2) = 2 × 6 × cos 35° = 9.83.
R = √96.63 = 9.83
tan β = B sin θ / (A + B cos θ) = 6 × sin 70° / (6 + 6 × cos 70°) = 0.7
β = 35° from A, towards B.

What the angle does

R for 6 and 6 at other angles

0° 120° from A
30° 11.5915° from A
45° 11.0922.5° from A
60° 10.3930° from A
90° 8.4945° from A
120° 660° from A
150° 3.1175° from A
180° 039.6° from A

Derivation

Why it works

With A = B the general result R² = A² + B² + 2AB cos θ becomes R² = 2A²(1 + cos θ).
Use 1 + cos θ = 2 cos²(θ/2) to get R² = 4A² cos²(θ/2), so R = 2A cos(θ/2).
The parallelogram is a rhombus, and a rhombus diagonal bisects the angle between its sides.
So R makes the angle β = θ/2 with each of the two vectors.

Questions