Vector Laws: Addition, Subtraction, Triangle & Parallelogram Law with Practice

1. The Triangle Law of Vector Addition

If two vectors A and B are represented by two sides of a triangle taken in order, then the third side of that triangle (taken in the opposite order) represents their resultant vector R = A + B.

Geometric steps:

1. Draw vector A from an initial point.
2. From the head of vector A, draw vector B.
3. The resultant vector R is drawn from the tail of A to the head of B.

Magnitude of resultant:
\[ R = \sqrt{A^2 + B^2 + 2AB\cos\theta} \] where \(\theta\) is the angle between vectors \(A\) and \(B\).

Example Numerics

Given: A = 5 units, B = 8 units, angle between vectors \(\theta = 60^\circ\).

Calculate the magnitude of the resultant vector.

R = √(5² + 8² + 2×5×8×cos60°)
  = √(25 + 64 + 80×0.5)
  = √(129)
  ≈ 11.36 units
  

2. The Parallelogram Law of Vector Addition

If two vectors A and B acting at a point are represented by two adjacent sides of a parallelogram, then the diagonal of the parallelogram drawn from that point represents the resultant vector R = A + B.

Geometric steps:

1. Draw vectors A and B with the same initial point.
2. Complete the parallelogram with these vectors as adjacent sides.
3. The diagonal from the shared initial point is the resultant vector.

Magnitude of resultant:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \] where \(\theta\) is the angle between \(A\) and \(B\).

Note: This formula is the same as in the triangle law because the triangle formed by adjoining vectors corresponds to half the parallelogram.

3. Vector Subtraction

Subtraction of vectors is done by adding the negative of one vector:

\[ A - B = A + (-B) \]

Where \(-B\) is a vector equal in magnitude to \(B\) but opposite in direction.

Numerical Example

Given vectors: \(A = 7\) units at \(0^\circ\), \(B = 5\) units at \(90^\circ\). Find the magnitude of \(A - B\).

Angle between A and -B = 180° - 90° = 90°

Magnitude of difference:
R = √(7² + 5² + 2×7×5×cos90°)
  = √(49 + 25 + 0)
  = √74
  ≈ 8.6 units
  

4. Practice Questions

1. State the triangle law of vector addition.
2. Describe the parallelogram law of vector addition.
3. Calculate the magnitude of resultant of vectors 6 units and 8 units at \(90^\circ\) using triangle/parallelogram law.
4. Given \(\mathbf{A} = 3\hat{i} + 2\hat{j}\), \(\mathbf{B} = -\hat{i} + 4\hat{j}\), find \(\mathbf{A}+\mathbf{B}\) and \(|\mathbf{A}+\mathbf{B}|\).
5. Two vectors of equal magnitude 10 units are inclined at \(120^\circ\). Find magnitude of resultant.
6. True/False: The vector addition is commutative (\(A + B = B + A\)).
7. If vectors \(\mathbf{A}\) and \(\mathbf{B}\) are opposite in direction and have magnitude 5 units each, what is magnitude of \(\mathbf{A} + \mathbf{B}\)?
8. Explain how vector subtraction is represented geometrically using the triangle law.
9. Using parallelogram law, calculate the resultant if \(\mathbf{A

   Vector Laws: Addition, Subtraction, Triangle & Parallelogram Law with Practice

   1. Triangle Law of Vector Addition

   If two vectors A and B are represented by two sides of a triangle taken in order, then the third side represents their resultant R = A + B.

   1. Draw vector A.
   2. From head of A, draw vector B.
   3. R is drawn from tail of A to head of B.

   Magnitude of resultant:
   \[ R = \sqrt{A^2 + B^2 + 2AB\cos\theta} \] where \(\theta\) is angle between vectors.

   Example

   A=5 units, B=8 units, \(\theta=60^\circ\). Find magnitude \(R\).

   R = √(5² + 8² + 2×5×8×cos 60°)
     = √(25 + 64 + 80×0.5)
     = √129 ≈ 11.36 units
     

   2. Parallelogram Law of Vector Addition

   If vectors A and B act at a point, placed tail-to-tail, completing the parallelogram, the diagonal from the tail is the resultant R = A + B.

   Magnitude:
   \[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]

   This matches the triangle law result.

   3. Vector Subtraction

   Subtraction: A - B = A + (-B), where \(-B\) has magnitude of \(B\) but opposite direction.

   Example

   A=7 units at \(0^\circ\), B=5 units at \(90^\circ\). Find magnitude of \(A - B\).

   Angle between A and -B = 180° - 90° = 90°
   
   |A - B| = √(7² + 5² + 2×7×5×cos 90°)
           = √(49 + 25 + 0)
           = √74 ≈ 8.6 units
     

   4. Practice Questions

   1. State the triangle law of vector addition.
   2. Describe the parallelogram law of vector addition.
   3. Calculate magnitude of resultant of vectors 6 and 8 units at \(90^\circ\).
   4. Find \( \mathbf{A} + \mathbf{B} \) and magnitude if \(\mathbf{A} = 3\hat{i} + 2\hat{j}\) and \(\mathbf{B} = -\hat{i} + 4\hat{j}\).
   5. Two vectors 10 units each inclined at \(120^\circ\). Find resultant magnitude.
   6. True/False: Vector addition is commutative, i.e., \(A + B = B + A\).
   7. If \(\mathbf{A}\) and \(\mathbf{B}\) have same magnitude 5 units but opposite directions, what is \(|\mathbf{A} + \mathbf{B}|\)?
   8. Explain geometrically how vector subtraction uses the triangle law.
   9. Using parallelogram law, find resultant of vectors 4 units and 5 units at \(60^\circ\).
   10. Given vectors \(\mathbf{a} = \langle 1, -2, 3 \rangle\) and \(\mathbf{b} = \langle 4, 0, -1 \rangle\), find \(\mathbf{a} + \mathbf{b}\) and its magnitude.

   5. Important Formulas Summary

   Concept	Formula
   Resultant magnitude (two vectors \(A,B\), angle \(\theta\))	\[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]
   Vector subtraction magnitude (equal magnitude \(a\), angle \(\theta\))	\[ |a - b| = \sqrt{2a^2(1 - \cos \theta)} = 2a \sin \frac{\theta}{2} \]
   Resultant direction angle \(\alpha\)	\[ \tan \alpha = \frac{B \sin \theta}{A + B \cos \theta} \]
   Component form addition	\(\mathbf{A} + \mathbf{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}\)