How to Solve Vector Addition Problems Using the Parallelogram Law
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
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Draw vectors P and Q from the same point.
Suppose P = 5 units, Q = 10 units, angle θ = 60°.
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Complete the parallelogram by drawing lines equal and parallel to P and Q.
This forms a four-sided figure with both vectors as sides.
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Draw the diagonal starting at the point where both vectors meet.
This diagonal is the resultant vector R.
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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: Definition, Formula & Applications
1. What is the Parallelogram Law of Vector Addition?
The Parallelogram Law of Vector Addition states that if two vectors acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant vector is represented in both magnitude and direction by the diagonal of the parallelogram passing through that same point. This is a geometric method for adding vectors.
2. What is the formula to find the magnitude of the resultant vector in the Parallelogram Law?
The magnitude of the resultant vector (R) of two vectors P and Q, inclined at an angle θ, is given by: |R| = √(P² + Q² + 2PQ cosθ). This formula is derived using the Law of Cosines.
3. How does the Parallelogram Law differ from the Triangle Law of Vector Addition?
Both laws achieve the same result—finding the resultant vector—but differ in their graphical representation. The Parallelogram Law uses a parallelogram with vectors tail-to-tail, while the Triangle Law arranges vectors head-to-tail. The resultant in the parallelogram law is the diagonal, while in the triangle law, it's the closing side of the triangle.
4. What do the two diagonals of the parallelogram represent in vector addition?
The diagonal originating from the common point of vectors P and Q represents their sum (P + Q, the resultant vector). The other diagonal, connecting the heads of P and Q, represents their difference (P - Q or Q - P).
5. Can the Parallelogram Law be used for vector subtraction?
Yes. Subtracting vector Q from vector P (P - Q) is equivalent to adding P and the negative of Q (P + (-Q)). Construct a parallelogram using P and -Q (opposite direction to Q). The diagonal represents the resultant of the subtraction.
6. How is the Parallelogram Law applied to physical quantities like forces?
In physics, if two forces, F1 and F2, act on a point, their resultant (net force) and direction are determined using the Parallelogram Law. The diagonal of the parallelogram formed by F1 and F2 represents the net force.
7. What is the main condition for applying the Parallelogram Law of vectors?
The vectors must be co-initial; their tails must originate from the same point. The law applies to vectors acting simultaneously on a single point.
8. What are some common applications of the Parallelogram Law?
The Parallelogram Law is used in various fields, including:
- Physics: Calculating the resultant of forces or velocities.
- Engineering: Analyzing stress and strain in structures.
- Mathematics: Understanding vector addition and properties in vector spaces.
9. How is the angle between the vectors related to the magnitude of the resultant?
The angle θ between vectors P and Q directly affects the magnitude of the resultant vector R. When θ = 0° (parallel vectors), R = P + Q. When θ = 180° (opposite directions), R = |P - Q|. When θ = 90° (perpendicular vectors), R = √(P² + Q²).
10. What is the significance of the parallelogram law in understanding vector components?
The Parallelogram Law helps visualize and calculate the resultant vector by resolving vectors into their components. It shows how individual components contribute to the overall magnitude and direction of the resultant, offering a deeper understanding of vector behavior.
11. Can the parallelogram law be applied to more than two vectors?
No, the parallelogram law, in its basic form, is specifically designed for adding two vectors. To add more than two vectors, you can use methods like the polygon law or resolving vectors into components and adding them algebraically.
12. What are some common mistakes students make when applying the Parallelogram Law?
Common errors include:
- Incorrectly measuring or drawing the vectors to scale.
- Misinterpreting the angle θ between the vectors.
- Not using the correct formula for calculating the magnitude and direction of the resultant vector.
- Forgetting that the resultant vector is a diagonal of the parallelogram.
