Addition of Vectors

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Addition of Vectors (Triangle Rule) | Eduvigyan

When two effects act together (like wind + river current), the final motion is not found by simple addition. We use vector addition to find the resultant (final speed and direction).

Topic: Vectors • Core idea: Resultant of two vectors at an angle

Boat–Wind–Current Story (Motivation)

Imagine you are sailing a boat in a river:

  • Wind pushes the boat in one direction.
  • Water current pushes your boat in another direction.

These two effects act together at an angle. So the natural questions are:

  • Where will you end up?
  • What will be the final direction?
  • What will be the final speed?
Boat in a river: wind vector and current vector giving resultant velocity
Figure 1: Boat–wind–current idea: two velocity vectors act together to produce a resultant (final) velocity.
To solve this “mystery”, we use the Triangle Rule of Vector Addition (also called the tip-to-tail method).

Triangle Rule of Vector Addition (Concept)

Suppose we have two vectors A and B, with an angle θ between them. We want their resultant vector R.

Step-by-step (Tip-to-tail)

  • Draw vector A.
  • From the tip (head) of A, draw vector B.
  • The vector from the tail of A to the tip of B is the resultant R.
Triangle rule of vector addition: tip-to-tail method with resultant R
Figure 2: Triangle rule (tip-to-tail): place B from the tip of A; the joining vector gives resultant R.
This resultant represents the combined effect — like wind + current or two forces acting together.

Resultant Magnitude (Derivation: clean & slow)

Consider the triangle formed by vector addition:

  • AB = A
  • BC = B
  • AC = R (resultant)

Drop a perpendicular from point C to the line AB meeting at M. Then:

Triangle ABC with AB=A, BC=B, AC=R and perpendicular CM to AB at M
Figure 3: Derivation diagram: drop a perpendicular to resolve B into components along and perpendicular to A.
AM = A + B cosθ CM = B sinθ Using Pythagoras: R² = AM² + CM² = (A + B cosθ)² + (B sinθ)² Expand: R² = A² + 2AB cosθ + B² cos²θ + B² sin²θ Since: cos²θ + sin²θ = 1 So: R² = A² + B² + 2AB cosθ Therefore: R = √(A² + B² + 2AB cosθ)
✅ This is the resultant magnitude formula for two vectors making angle θ.

Quick Understanding (Special Angles)

Angle between A and B cosθ Resultant magnitude Meaning
θ = 0° 1 R = A + B Same direction → maximum resultant
θ = 90° 0 R = √(A² + B²) Perpendicular → Pythagoras
θ = 180° -1 R = |A − B| Opposite direction → subtraction
Common mistake: Students often add magnitudes directly even when vectors are at an angle. That is incorrect unless they are along the same straight line.

Resultant Direction (Angle α)

Let α be the angle between the resultant R and vector A. From the same right-triangle:

tan α = CM / AM = (B sinθ) / (A + B cosθ) So: α = tan⁻¹( (B sinθ) / (A + B cosθ) )
✅ This gives the direction of the resultant relative to vector A.

More Important Ideas in Addition of Vectors

1) Parallelogram Law

If two vectors start from the same point, draw them as adjacent sides of a parallelogram. The diagonal from the starting point gives the resultant.

Parallelogram law of vector addition with diagonal as resultant
Figure 5: Parallelogram law: the diagonal represents the same resultant as the triangle rule.
Triangle rule and parallelogram law are the same idea, just drawn differently.

2) Head-to-tail method for many vectors

For adding 3 or more vectors, place them head-to-tail in sequence. The resultant connects the start of the first to the end of the last.

Head-to-tail addition of multiple vectors (polygon method) and resultant
Figure 6: Polygon method: add many vectors by placing each next vector from the tip of the previous one.

3) Components method (powerful alternative)

If a vector makes an angle φ with the x-axis, then:

A vector resolved into x and y components Ax and Ay
Figure 7: Components method: resolve vectors into x- and y-components and add them separately.
Ax = A cosφ Ay = A sinφ Rx = Ax + Bx Ry = Ay + By R = √(Rx² + Ry²) Direction: tan α = Ry / Rx
Components method becomes the easiest approach when you have many vectors at different angles.

Solved Numerical Examples

Example 1: Wind + current (perpendicular)

A boat moves at 4 m/s due to wind towards east, and the river current flows at 3 m/s towards north. Find the resultant speed and direction.

Solution:

Here A = 4 (east), B = 3 (north), and θ = 90°.

R = √(A² + B²) = √(4² + 3²) = √25 = 5 m/s tan α = B/A = 3/4 α = tan⁻¹(3/4) ≈ 36.87° (north of east)

Resultant speed = 5 m/s, direction = 36.9° north of east.

Example 2: Two forces at an angle

Two forces of 10 N and 6 N act at an angle of 60°. Find the magnitude of the resultant force.

Solution:
R = √(A² + B² + 2AB cosθ) = √(10² + 6² + 2×10×6×cos60°) = √(100 + 36 + 120×0.5) = √196 = 14 N

Resultant force magnitude = 14 N.

Example 3: Opposite directions (subtraction)

Two vectors 18 N east and 11 N west act on a body. Find the resultant (magnitude + direction).

Solution:
Take east as +, west as −: R = +18 + (−11) = +7 N

Resultant = 7 N towards east.

Example 4: General case with angle

Two vectors have magnitudes A = 8 units and B = 5 units with angle θ = 53° between them. Find (i) resultant magnitude and (ii) angle α with vector A. (Use cos53≈0.6, sin53≈0.8)

Solution:
R = √(A² + B² + 2AB cosθ) = √(8² + 5² + 2×8×5×0.6) = √(64 + 25 + 48) = √137 ≈ 11.70 tan α = (B sinθ) / (A + B cosθ) = (5×0.8) / (8 + 5×0.6) = 4 / 11 α ≈ tan⁻¹(0.3636) ≈ 20°

Resultant ≈ 11.7 units, direction ≈ 20° from A towards B.

Practice Questions (For Students)

A) Conceptual Questions

  1. State the triangle rule of vector addition in your own words.
  2. Why can’t we add magnitudes directly when vectors are at an angle?
  3. What is the difference between the triangle rule and the parallelogram law?
  4. For which angle between vectors is the resultant maximum? For which is it minimum?
  5. If A + B = 0, what must be true about A and B?

B) Numerical Questions

  1. Two vectors of magnitudes 9 and 12 act at 90°. Find the resultant magnitude.
  2. A boat has speed 6 m/s towards east. River current is 8 m/s towards north. Find resultant speed and direction.
  3. Two forces 15 N and 10 N act at an angle of 120°. Find the magnitude of resultant force.
  4. Two displacements 20 m and 15 m act at 60°. Find the magnitude of resultant displacement.
  5. Using components: A = 10 units at 0° and B = 10 units at 90°. Find R and its angle with +x axis.
  6. Two vectors 7 units and 9 units act at 30°. Find the resultant magnitude (use cos30 = 0.866).
Answers (Click to reveal)
1) √(9² + 12²) = 15 2) R = √(6² + 8²) = 10 m/s α = tan⁻¹(8/6) ≈ 53.13° north of east 3) cos120 = −1/2 R = √(15² + 10² + 2×15×10×(−1/2)) = √175 ≈ 13.23 N 4) cos60 = 1/2 R = √(20² + 15² + 2×20×15×1/2) = √925 ≈ 30.41 m 5) Rx = 10, Ry = 10 R = √(10² + 10²) ≈ 14.14, angle = 45° 6) R = √(7² + 9² + 2×7×9×cos30) = √(49 + 81 + 126×0.866) = √(239.116) ≈ 15.46

Summary (Takeaway)

  • Vector addition finds the resultant of two or more vectors.
  • Triangle rule (tip-to-tail) gives both magnitude and direction of the resultant.
  • For two vectors at angle θ:
    R = √(A² + B² + 2AB cosθ) α = tan⁻¹( (B sinθ) / (A + B cosθ) )
  • Special angles (0°, 90°, 180°) help you quickly understand max/min cases.
  • For multiple vectors, the components method is the most powerful tool.
Next topic: Resolution of Vectors (Components) and Unit Vectors (i, j, k).
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