Numford
Vectors

When is the resultant maximum and minimum?

The resultant is largest, 11, when the vectors point the same way, and smallest, 3, when they point opposite ways.

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 between0°cos θ = A·B / AB
Unit vectors1iB = 1i

Vector A of size 7 and vector B of size 4 with 0 degrees between them. The resultant is 11, 0 degrees from A. The dot product is 28 and the cross product has size 0.

Show the formulas
Resultant
R = √(A² + B² + 2AB cos θ) = √(7² + 4² + 2 × 7 × 4 × cos 0°) = 11
Direction
tan β = B sin θ / (A + B cos θ) = 4 × sin 0° / (7 + 4 × cos 0°), so β = 0° from A
Difference
|A - B| = √(A² + B² - 2AB cos θ) = √(7² + 4² - 2 × 7 × 4 × cos 0°) = 3
Dot product
A·B = AB cos θ = 7 × 4 × cos 0° = 28
Cross product
|A×B| = AB sin θ = 7 × 4 × sin 0° = 0
Components
A·B = AxBx + AyBy + AzBz = 7 × 4 + 0 × 0 + 0 × 0 = 28
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 = 7 and B = 4. Only θ can change.
  2. Largest: cos θ = 1 at θ = 0°, so R = A + B = 7 + 4 = 11.
  3. Smallest: cos θ = -1 at θ = 180°, so R = |A - B| = |7 - 4| = 3.
  4. At any other angle 3 ≤ R ≤ 11. If A = B the minimum is zero.
What the angle does

R for 7 and 4 at other angles

  • 0° 110° from A
  • 30° 10.6510.82° from A
  • 45° 10.2316.05° from A
  • 60° 9.6421.05° from A
  • 90° 8.0629.74° from A
  • 120° 6.0834.72° from A
  • 150° 4.0629.49° from A
  • 180° 30° from A
Derivation

Why it works

  1. The resultant is R = √(A² + B² + 2AB cos θ), and only cos θ changes with the angle.
  2. cos θ is largest at θ = 0° where it is +1, so R is largest: R = √(A + B)² = A + B.
  3. cos θ is smallest at θ = 180° where it is -1, so R is smallest: R = √(A - B)² = |A - B|.
  4. Every other angle gives a value between the two, so |A - B| ≤ R ≤ A + B.
  5. This is also the triangle rule: one side of a triangle is never longer than the other two together, and never shorter than their difference.
Questions

Maximum and minimum resultant: quick answers

When is the resultant of two vectors maximum?

When they point the same way (θ = 0 degrees). Then R = A + B = 11 for 7 and 4.

When is the resultant minimum?

When they point opposite ways (θ = 180 degrees). Then R = |A - B| = 3, and it is zero if the magnitudes are equal.

Can the resultant be larger than A + B?

No. By the triangle rule one side is never longer than the other two together, so |A - B| ≤ R ≤ A + B.

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