Vectors addition | Parallelogram law

Eleven Standard >> Vectors addition | Parallelogram law

Mastering Vector Addition Using the Parallelogram Law

Vector addition is a fundamental concept in physics and mathematics, essential for combining forces, velocities, and displacements. While the triangle law focuses on sequential tip-to-tail addition, the Parallelogram Law offers a geometric method to find the resultant of two vectors acting simultaneously at a point. This article explains the parallelogram law, its derivation, applications, and practical steps to solve problems effectively.

1. What is the Parallelogram Law of Vector Addition?

The Parallelogram Law states that if two vectors are represented as adjacent sides of a parallelogram, their resultant vector is the diagonal of the parallelogram drawn from the same starting point.

Formula:

If vectors \( \vec{A} \) and \( \vec{B} \) have an angle \( \theta \) between them, then:

\( \vec{R} = \vec{A} + \vec{B} \)

Magnitude:

\( |vec{R}| = \sqrt{A^2 + B^2 + 2AB\cos\theta} \)

Direction:

\( phi = \tan^{-1} \left( \frac{B\sin\theta}{A + B\cos\theta} \right) \)

Here, \( \phi \) is the angle between \( \vec{R} \) and \( \vec{A} \).

2. Step-by-Step Procedure to Add Vectors Using the Parallelogram Law

1. Draw Vectors Tail-to-Tail: Represent \( \vec{A} \) and \( \vec{B} \) as adjacent sides starting from the same origin.
2. Complete the Parallelogram: Draw sides parallel to each vector to form a parallelogram.
3. Draw the Resultant Vector: The diagonal from the common origin to the opposite corner gives \( \vec{R} \).
4. Calculate: Use cosine law for magnitude and trigonometry for direction.

3. Visualizing the Parallelogram Law

Diagram: Vectors \( \vec{A} \) and \( \vec{B} \) form adjacent sides of a parallelogram. The diagonal is \( \vec{R} \).

Example: Two tugboats apply 1000 N and 800 N forces at 50° to each other. Resultant force determines the ship’s net acceleration.

4. Special Cases

• Perpendicular Vectors (\( \theta = 90^{\circ} \)):
  \( R = \sqrt{A^2 + B^2} \)
  Example: 3 m/s east and 4 m/s north → 5 m/s northeast.
• Collinear Vectors (\(\theta = 0^{\circ} \) or \( 180^{\circ} \)):
  Same direction: \( R = A + B \)
  Opposite direction: \( R = |A - B| \)

5. Applications of the Parallelogram Law

• Mechanics: Analyzing forces in equilibrium.
• Navigation: Net velocity of aircraft with wind.
• Engineering: Resolving structural stresses.
• Electromagnetism: Superposing field vectors.

6. Common Mistakes to Avoid

• Incorrect angle: \( \theta \) is measured between tail-to-tail vectors.
• Mixing up laws: Parallelogram law is not tip-to-tail.
• Ignoring direction: Always apply angle and direction correctly.

Pro Tip: Label all vectors, angles, and diagonals clearly in your diagram!

7. Frequently Asked Questions (FAQs)

Q1. Can the parallelogram law add more than two vectors?

A: No. It applies to two vectors. For more, use polygon or component method.

Q2. How is this different from the triangle law?

A: Triangle law uses tip-to-tail addition. Parallelogram law uses tail-to-tail addition and a diagonal.

Q3. How to subtract vectors using the parallelogram law?

A: Use the negative vector: \( vec{A} - \vec{B} = \vec{A} + (-\vec{B}) \). Draw \( -\vec{B} \) in the opposite direction to \( \vec{B} \).

8. Practice Problem

Problem: Two forces of 10 N and 15 N act at 60° to each other. Find the resultant's magnitude and direction.

Solution:

\( R = \sqrt{10^2 + 15^2 + 2(10)(15)\cos 60^\circ} = \sqrt{100 + 225 + 150} = \sqrt{475} \approx 21.8 , \text{N} \)

\( \phi = \tan^{-1} \left( \frac{15\sin 60^{\circ}}{10 + 15\cos 60^{\circ}} \right) \approx 36.6^{\circ} \)