What is the Parallelogram Law Formula and How to Use It
The concept of Parallelogram Law is essential in mathematics, especially for understanding how vectors are added both graphically and algebraically. Mastering the parallelogram law helps students solve physics and maths problems quickly and appears in many board and competitive exam questions.
What Is Parallelogram Law?
The Parallelogram Law states that if two vectors are drawn from a common point, they can be represented as the adjacent sides of a parallelogram, and their sum (resultant) is shown by the diagonal of the parallelogram starting at the same point. This law appears in topics like Vector Algebra, the addition of forces, and geometric proofs.
Key Formula for Parallelogram Law
Here’s the standard formula: \( R = \sqrt{P^2 + Q^2 + 2PQ\cos\theta} \)
Where P and Q are the magnitudes of the two vectors and θ is the angle between them.
Cross-Disciplinary Usage
Parallelogram Law is used not only in Maths, but is crucial in Physics to find resultant forces, in engineering statics, and even in computer science graphics for resolving directions. For JEE, NEET, and school exams, it explains how real forces or motions combine in real life.
Step-by-Step Illustration
-
Draw vectors P and Q from the same point.
Suppose P = 5 units, Q = 10 units, angle θ = 60°.
-
Complete the parallelogram by drawing lines equal and parallel to P and Q.
This forms a four-sided figure with both vectors as sides.
-
Draw the diagonal starting at the point where both vectors meet.
This diagonal is the resultant vector R.
-
Use the formula:
\( R = \sqrt{5^2 + 10^2 + 2 \times 5 \times 10 \times \cos 60^\circ} \)
\( R = \sqrt{25 + 100 + 100 \times 0.5} = \sqrt{25 + 100 + 50} = \sqrt{175} \approx 13.23 \) units
- So, the resultant vector’s magnitude is approximately 13.23.
Speed Trick or Vedic Shortcut
Here’s how to quickly estimate the resultant using the parallelogram law if vectors are at 90° (right angles): Add the squares, then find the square root—just like Pythagoras’ theorem!
Example: For two vectors, 8 units and 6 units perpendicular to each other:
- Resultant = \( \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \) units
This trick is very useful in MCQs and quick calculations. Vedantu’s live classes also teach such shortcuts to help you solve faster.
Try These Yourself
- Two vectors of 7 N and 24 N act at 90°. Find the resultant using the parallelogram law.
- If the angle between vectors is 0°, what will the resultant be?
- Explain the difference between the triangle law and parallelogram law of vector addition. (Hint: See the difference here)
- Draw a vector diagram for P = 3 units, Q = 4 units, angle = 60°, and find R.
Frequent Errors and Misunderstandings
- Forgetting to use the cosine of the angle (especially when not perpendicular or parallel).
- Mixing up when to use triangle law and when to use parallelogram law.
- Not drawing vectors from the same starting point—co-initial vectors are required.
Relation to Other Concepts
The Parallelogram Law connects closely to the Vector Algebra, and the Properties of Vectors. Understanding it also helps with Vector Cross Product and more advanced geometry and physics problems.
Classroom Tip
A helpful way to remember the parallelogram law: Draw both vectors from a common point, complete the parallelogram, then the diagonal through that point is always the sum. Vedantu teachers often use simple chalk diagrams and arrow notations to visualize this and build confidence among students.
We explored Parallelogram Law—from definition, formula, to quick solved examples and common mistakes. Practising problems and discussing them with Vedantu’s expert teachers will help you apply this concept with confidence in both exams and real-world situations.
Vector Algebra | Vector Cross Product
FAQs on Parallelogram Law of Vector Addition
1. What is the Parallelogram Law?
The Parallelogram Law states that the sum of the squares of the diagonals of a parallelogram equals the sum of the squares of all four sides. In geometry, it is written as:
AC² + BD² = 2(AB² + BC²)
In vector form, it states that for two vectors a and b:
|a + b|² + |a − b|² = 2(|a|² + |b|²)
This law is widely used in coordinate geometry, vector algebra, and physics.
2. What is the formula for the Parallelogram Law in vectors?
The vector form of the Parallelogram Law is |a + b|² + |a − b|² = 2(|a|² + |b|²). It relates the magnitudes of two vectors and their sum and difference.
- |a + b| represents one diagonal of the parallelogram.
- |a − b| represents the other diagonal.
- |a| and |b| are the magnitudes of the sides.
3. How do you prove the Parallelogram Law?
The Parallelogram Law is proved using vector algebra and dot product properties. Start with:
- |a + b|² = (a + b) · (a + b)
- |a − b|² = (a − b) · (a − b)
- = a·a + 2a·b + b·b + a·a − 2a·b + b·b
- = 2a·a + 2b·b
- = 2(|a|² + |b|²)
4. Can you give an example of the Parallelogram Law?
Yes, for vectors a = (1,2) and b = (3,4), the Parallelogram Law can be verified numerically.
- |a|² = 1² + 2² = 5
- |b|² = 3² + 4² = 25
- 2(|a|² + |b|²) = 2(5 + 25) = 60
- a + b = (4,6) → |a + b|² = 16 + 36 = 52
- a − b = (−2,−2) → |a − b|² = 4 + 4 = 8
- 52 + 8 = 60
5. What is the Parallelogram Law in geometry?
In geometry, the Parallelogram Law states that the sum of the squares of the diagonals equals twice the sum of the squares of adjacent sides. If ABCD is a parallelogram:
AC² + BD² = 2(AB² + BC²)
This property helps in solving problems related to diagonals, side lengths, and coordinate geometry.
6. How is the Parallelogram Law used in vector addition?
The Parallelogram Law of vector addition states that the sum of two vectors is represented by the diagonal of the parallelogram formed by them.
- Draw vectors a and b from the same initial point.
- Complete the parallelogram.
- The diagonal represents a + b.
7. What is the difference between the Parallelogram Law and the Triangle Law?
The key difference is that the Parallelogram Law uses two vectors from a common point, while the Triangle Law places vectors head-to-tail.
- Parallelogram Law: Resultant is the diagonal of a parallelogram.
- Triangle Law: Resultant is the third side of a triangle.
8. Why is the Parallelogram Law important in mathematics?
The Parallelogram Law is important because it characterizes inner product spaces and helps define vector magnitude and orthogonality. It is used in:
- Vector algebra
- Coordinate geometry
- Physics (force and velocity addition)
- Hilbert spaces in higher mathematics
9. Does the Parallelogram Law apply to all quadrilaterals?
No, the Parallelogram Law applies only to parallelograms, not all quadrilaterals. The property AC² + BD² = 2(AB² + BC²) holds only when opposite sides are parallel and equal. It does not generally hold for rectangles, trapeziums, or irregular quadrilaterals unless they satisfy parallelogram conditions.
10. How do you derive the Parallelogram Law using dot product?
The Parallelogram Law is derived using dot product identities of vectors.
- |a + b|² = a·a + 2a·b + b·b
- |a − b|² = a·a − 2a·b + b·b
- Add both equations to get 2a·a + 2b·b
- Which equals 2(|a|² + |b|²)
