The parallelogram law of vector addition provides a method for finding the resultant vector when two vectors are combined.
Understanding the Parallelogram Law
Based on the reference, the parallelogram law of vector addition states that:
- If two vectors are considered to be the two adjacent sides of a parallelogram.
- Their tails must meet at a common point.
- Then, the diagonal of the parallelogram originating from this common point will be the resultant vector.
In simpler terms, if you have two vectors, let's call them Vector A and Vector B, starting from the same origin, you can complete a parallelogram using these two vectors as sides. The diagonal drawn from the same origin where A and B start represents the single vector that has the same effect as combining Vector A and Vector B. This diagonal is the resultant vector (R = A + B).
This law is geometrically equivalent to the triangle law of vector addition; it just visualizes the process differently, showing the commutative property of vector addition (A + B = B + A).
Deriving the Magnitude of the Resultant Vector
When we add two vectors, we are often interested in the size or length of the resultant vector, which is called its magnitude. We can derive a general expression for the magnitude of the resultant vector using trigonometry.
Let's consider two vectors, A and B, whose tails are joined at a common point. Let the angle between A and B be θ.
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Form the Parallelogram: Draw the vectors A and B originating from the same point O. Complete the parallelogram OACB, where OA represents A and OB represents B. The resultant vector R is represented by the diagonal OC.
(Note: The image is illustrative of the parallelogram setup) -
Consider the Triangle: Focus on the triangle OAC. The sides are OA (representing A), AC (which is parallel and equal in length to OB, representing B), and OC (representing R).
- Length OA = |A| = A
- Length AC = |B| = B
- Length OC = |R| = R
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Find the Angle: The angle between vectors A and B at O is θ (angle AOB). In the parallelogram OACB, the angle between sides OA and AC inside triangle OAC is not θ. The angle at O in triangle OAC is angle AOC, which is not θ. However, we can use the angle between OA and an extension of OA, or the angle at C.
Alternatively, we can use the angle opposite the resultant vector R in triangle OAB (where AB is the other diagonal, not the resultant). More commonly, we extend one vector and drop a perpendicular to form a right-angled triangle to apply the Pythagorean theorem, or directly apply the Cosine Rule to triangle OAC. -
Using the Cosine Rule: Let's apply the Cosine Rule to triangle OAC. The angle at A inside the triangle OAC is adjacent to angle θ. Since OA is parallel to BC and OB is parallel to AC, OACB is a parallelogram. The angle at A in triangle OAC is the angle between OA and AC. The angle between the lines OA and OB is θ. In the parallelogram, the adjacent angles sum to 180°. So, the angle BAC = 180° - θ. The angle OAC within triangle OAC is equal to angle AOB, which is θ (by parallel lines properties or considering the other angle in the parallelogram). Correction: The angle opposite the side OC (the resultant R) in triangle OAC is the angle OAC. The angle AOB is θ. The angle OAC is not θ.
Let's use the angle between A and B, which is θ. The adjacent angle in the parallelogram to θ is 180° - θ. This angle is at point B and point A in triangle OAB (where OAB is the upper triangle, not the resultant diagonal).
For triangle OAC, the angle opposite the side OC (R) is the angle at A, which is angle OAC. In parallelogram OACB, angle AOB = θ. The angle at A inside the triangle OAC is adjacent to the angle θ from vector B extended. Consider extending vector A past O. The angle between extended A and B is 180° - θ. The internal angle of the parallelogram at A (angle CAB) is 180° - θ. Thus, the angle opposite the resultant R in triangle OAB (if we used that diagonal) would be θ, and the side AB would be the resultant of A - B or B - A.Let's reconsider the diagram OACB where OC is the resultant R. The angle between OA (A) and OB (B) is θ. In triangle OAC, side OA = A, side AC = B, and side OC = R. The angle opposite side R is the angle OAC. The angle adjacent to θ at O is 180° - θ. Consider triangle OAB where AB is the other diagonal. Using the cosine rule on triangle OAC with angle OAC is difficult as OAC is not simply related to θ.
A clearer derivation uses the cosine rule on triangle OAC where the angle opposite the side OC (the resultant R) is needed. Let's form a right-angled triangle instead.
Extend OA to point P. Draw CD perpendicular to OP.- In right-angled triangle ADC: AC = B. Angle CAP = θ (corresponding to angle AOB).
- AD = AC cos(θ) = B cos(θ)
- CD = AC sin(θ) = B sin(θ)
- Now consider the right-angled triangle ODC: OD = OA + AD = A + B cos(θ) and CD = B sin(θ).
- Using the Pythagorean theorem on triangle ODC:
R² = OC² = OD² + CD²
R² = (A + B cos(θ))² + (B sin(θ))²
R² = A² + 2AB cos(θ) + B² cos²(θ) + B² sin²(θ)
R² = A² + 2AB cos(θ) + B² (cos²(θ) + sin²(θ))
Since cos²(θ) + sin²(θ) = 1:
R² = A² + B² + 2AB cos(θ)
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Final Expression: Taking the square root of both sides gives the magnitude of the resultant vector R:
|R| = √(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 vector A and vector B when their tails are joined.
Summary of Resultant Magnitude Formula
| Term | Description |
|---|---|
| R | |
| A | Magnitude of the first vector ( |
| B | Magnitude of the second vector ( |
| θ (theta) | Angle between vector A and vector B |
Practical Examples & Special Cases
- Vectors in the same direction (θ = 0°): cos(0°) = 1. R = √(A² + B² + 2AB) = √((A + B)²) = A + B. The magnitudes simply add up.
- Vectors in opposite directions (θ = 180°): cos(180°) = -1. R = √(A² + B² - 2AB) = √((A - B)²) = |A - B|. The resultant magnitude is the difference of their magnitudes.
- Vectors perpendicular (θ = 90°): cos(90°) = 0. R = √(A² + B² + 0) = √(A² + B²). This is the Pythagorean theorem, as expected for perpendicular vectors.
This formula is fundamental in physics and engineering for combining forces, velocities, or any other vector quantities.
