Triangle Law of Vector Addition
The triangle law of vector addition states that if two vectors are represented by two sides of a triangle taken in the same order, then the third side of the triangle represents the resultant vector.
Understanding the Concept
Imagine you walk 5 meters east (vector A) and then 3 meters north (vector B). The triangle law helps us find the shortest distance between your starting point and your final position, which is represented by the resultant vector (vector R).
Visual Representation
In the diagram, the vectors A and B are represented by two sides of the triangle. The resultant vector R is represented by the third side, which is the diagonal of the parallelogram formed by vectors A and B.
Expression for the Magnitude of the Resultant Vector
The magnitude of the resultant vector (R) can be calculated using the law of cosines:
R² = A² + B² - 2AB cos θ
where:
- R is the magnitude of the resultant vector
- A is the magnitude of vector A
- B is the magnitude of vector B
- θ is the angle between vectors A and B
Example
Let's say vector A has a magnitude of 5 meters and vector B has a magnitude of 3 meters. The angle between them is 60 degrees.
Using the law of cosines:
R² = 5² + 3² - 2 5 3 * cos 60°
R² = 25 + 9 - 15
R² = 19
R = √19 ≈ 4.36 meters
Therefore, the magnitude of the resultant vector is approximately 4.36 meters.
Practical Insights
The triangle law of vector addition is widely used in various fields, including:
- Physics: Calculating the resultant force acting on an object.
- Engineering: Determining the resultant velocity of a moving object.
- Navigation: Finding the shortest distance between two points.
