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 does the Triangle Law state in the context of adding vectors?

According to the Triangle Law of Vector Addition, if two vectors are arranged such that the tail of the second begins at the head of the first, then the vector representing their sum is the side of the triangle that connects 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

Draw the First Vector \( (\vec{A} )\):
Draw vector \( \vec{A} \) as an arrow, ensuring its length reflects the size (magnitude) of the quantity it represents.

Example: A displacement of 4 units east.
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.
Resultant Vector (\( \vec{R} \)):
Place the starting point of ( vec{A} ) at the ending point of ( vec{B} ) to arrange them in tip-to-tail order.

\( \vec{R} \) represents the combined effect.
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 travels northward at 4 m/s while the river current flows eastward 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. This concept extends to the polygon law by connecting multiple vectors one after another in a tip-to-tail manner.

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

A: In the triangle method, vectors are added one after another, while in the parallelogram method, both vectors start from the same point.

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: After moving 8 kilometers eastward, the car takes a turn and proceeds 6 kilometers 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