Vectors addition | triangle law

Eleven Standard >> Vectors addition | triangle law

Mastering Vector Addition Using the Triangle Law

Vector addition is a cornerstone of physics and mathematics, enabling us to combine quantities like forces, velocities, and displacements. Among the methods for adding vectors, the Triangle Law is one of the most intuitive and widely used. This article breaks down the triangle law, its applications, step-by-step procedures, and common pitfalls to help you ace exams and solve real-world problems.

1. What is the Triangle Law of Vector Addition?

The Triangle Law states that if two vectors are represented as two sides of a triangle taken in tip-to-tail order, the resultant vector is given by the third side of the triangle drawn from the starting point of the first vector to the endpoint of the second.

Formula: If vectors \( \vec{A} \) and \( \vec{B} \) form two sides of a triangle, the resultant \( \vec{R} \) is:

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

Magnitude:

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

Direction:

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

Here, \( \theta \) is the angle between \( \vec{A} \) and \( \vec{B} \) when placed tip-to-tail.

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

1. Draw the First Vector \( (\vec{A} )\):

   Represent \( \vec{A} \) as an arrow with length proportional to its magnitude.

   Example: A displacement of 4 units east.

2. Draw the Second Vector (\( \vec{B} \)) Tip-to-Tail:

   Place the tail of \( \vec{B} \) at the tip of \( \vec{A} \).

   Example: A displacement of 3 units north.

3. Resultant Vector (\( \vec{R} \)):

   Connect the tail of \( \vec{A} \) to the tip of \( \vec{B} \).

   \( \vec{R} \) represents the combined effect.

4. Calculate Magnitude and Direction:

   For \( \theta = 90^{\circ} \):

   \( R = \sqrt{A^2 + B^2} \)

   For general angles: Use cosine and sine rules.

3. Visualizing the Triangle Law

Diagram: Two vectors \( \vec{A} \) and \( \vec{B} \) form a triangle, and \( \vec{R} \) is the closing side from the start of \( \vec{A} \) to the end of \( \vec{B} \).

Example: A boat moves north at 4 m/s in a river flowing east at 3 m/s. Resultant velocity = 5 m/s northeast.

4. Special Cases

• Collinear Vectors (Same Direction): \( R = A + B \)
• Collinear (Opposite Directions): \( R = |A - B| \)
• Perpendicular Vectors: \( \theta = 90^{\circ} \), so \( R = \sqrt{A^2 + B^2} \)

5. Applications of the Triangle Law

• Mechanics: Analyzing forces in equilibrium.
• Navigation: Resultant velocity of ships or aircraft.
• Engineering: Structural vector analysis.
• Electromagnetism: Net electric or magnetic fields.

6. Common Mistakes to Avoid

• Incorrect tip-to-tail placement: Always attach the tail of the second vector to the tip of the first.
• Ignoring direction: Measure \( \theta \) correctly between the two vectors.
• Confusing with parallelogram law: Triangle law is sequential addition; parallelogram law uses a common origin.

Pro Tip: Always draw a vector diagram—it helps avoid mistakes.

7. Frequently Asked Questions (FAQs)

Q1. Can the triangle law be used for more than two vectors?

A: Yes. Extend it to the polygon law by adding vectors tip-to-tail in sequence.

Q2. How is the triangle law different from the parallelogram law?

A: Triangle law adds vectors sequentially; parallelogram law adds from a common origin.

Q3. How to subtract vectors using the triangle law?

A: Add the negative vector: \( \vec{A} - \vec{B} = \vec{A} + (-\vec{B}) \)

8. Practice Problem

Problem: A car moves 8 km east and then 6 km northwest. Find the resultant displacement.

Angle between east and northwest = \( \theta = 135^{\circ} \)

\( R = \sqrt{8^2 + 6^2 + 2 \cdot 8 \cdot 6 \cdot \cos(135^{\circ})} \)

Answer: \( R \approx 5.2 \) km