Numford
Vectors

Resultant of two vectors at 180 degrees

For 8 and 5 at 180 degrees, R = 3, at 0° 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 between180°cos θ = A·B / AB
Unit vectors1iB = -1i

Vector A of size 8 and vector B of size 5 with 180 degrees between them. The resultant is 3, 0 degrees from A. The dot product is -40 and the cross product has size 0.

Show the formulas
Resultant
R = √(A² + B² + 2AB cos θ) = √(8² + 5² + 2 × 8 × 5 × cos 180°) = 3
Direction
tan β = B sin θ / (A + B cos θ) = 5 × sin 180° / (8 + 5 × cos 180°), so β = 0° from A
Difference
|A - B| = √(A² + B² - 2AB cos θ) = √(8² + 5² - 2 × 8 × 5 × cos 180°) = 13
Dot product
A·B = AB cos θ = 8 × 5 × cos 180° = -40
Cross product
|A×B| = AB sin θ = 8 × 5 × sin 180° = 0
Components
A·B = AxBx + AyBy + AzBz = 8 × -5 + 0 × 0 + 0 × 0 = -40
Determinant
A×B = (AyBz - AzBy) i + (AzBx - AxBz) j + (AxBy - AyBx) k = 0
Unit vector
 = A / |A| = 1i
Working

Step by step

  1. Given A = 8, B = 5 and the angle between them θ = 180°.
  2. R² = A² + B² + 2AB cos θ = 8² + 5² + 2 × 8 × 5 × cos 180° = 9
  3. cos 180° = -1, so R = |A - B|. The vectors oppose and the bigger one wins.
  4. R = √9 = 3
  5. tan β = B sin θ / (A + B cos θ) = 5 × sin 180° / (8 + 5 × cos 180°) = 0
  6. β = 0° from A, towards B.
What the angle does

R for 8 and 5 at other angles

  • 0° 130° from A
  • 30° 12.5811.46° from A
  • 45° 12.0717.04° from A
  • 60° 11.3622.41° from A
  • 90° 9.4332.01° from A
  • 120° 738.21° from A
  • 150° 4.4434.26° from A
  • 180° 30° from A
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 180 degrees: quick answers

What is the resultant of 8 and 5 at 180 degrees?

R = √(8² + 5² + 2 × 8 × 5 × cos 180°) = 3, making 0° with A.

What changes if the angle is not 180 degrees?

Only cos θ changes. The resultant is largest, 13, at 0 degrees and smallest, 3, 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.

Try your own vectors