Vectors - Derivation of Addition Formula

Description:

Formula to find the magnitude and direction of the resultant vector

Note

• Parallelogram law of addition is used for this derivation.
• Triangle Law method will yield the same result.

Derivation

a→  and b→  are considered as the original vectors. Angle between them is ‘θ’

S→ is the resultant vector.

Aim is to find the magnitude (length) of the resultant vector (|S|→) and the angle ‘α’ (angle between the resultant and a→).

Construction −

• Extend the line AD

• Drop a perpendicular from point C on to the extended line. Perpendicular meets the line at point B.

▱ AEDC is a parallelogram, hence AE = DC. Hence, DC  = |b|→ or DC = b.

▱ AEDC is a parallelogram, hence AE∥DC. Also, AD is a transversal to it. This implies ∠EAD = ∠CDB = θ.

Now, in △BCD,

• sin θ = BCDC, hence, BC = |b|→ sin θ or b sin θ

• cos θ = BDCD, hence, BD = |b|→ cos θ or b cos θ

Finally, in △ABC,

• Using Pythagoras’ theorem, |S|→² = (|a|→ + |b|→ cos θ)² + (|b|→ sin θ)²

• This implies, |S|→ = √(|a|→ + |b|→ cos θ)² + (|b|→ sin θ)²

This implies, |S|→ = √|a|→² + |b|→² + 2(|a|→)(|b|→)cos θ .. PROVED

Also, tan θ =

|b|→sin θ|a|→ + |b|→cos θ... PROVED

Note

1. The magnitude of resultant depends on two factors −

• The magnitudes of original vectors

• Angle between the original vectors

2. The sign convention for the angle is −

• Anticlockwise direction is positive (+)

• Clockwise direction is negative (-)