Vector Addition and Subtraction: Graphical Methods
1. Adding Vectors
- Head-to-Tail (Triangle) Method:
- Place the tail of vector B at the head of vector A.
- Draw a line from the tail of A to the head of B. This line is the resultant R = A + B.
- Parallelogram Method:
- Bring both vectors A and B to a common origin.
- Draw a parallelogram. The diagonal from the origin is the resultant R = A + B.
- Commutative Property:
\( \mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A} \)
2. Subtracting Vectors
- Subtraction is addition with the opposite vector:
\( \mathbf{A} – \mathbf{B} = \mathbf{A} + (-\mathbf{B}) \) - Example: To find A − B, add A and −B using the head-to-tail method.
3. Null Vector (Zero Vector)
- Result of adding a vector and its negative:
\( \mathbf{A} – \mathbf{A} = \mathbf{0} \) - Properties:
- \( \mathbf{A} + \mathbf{0} = \mathbf{A} \)
- \( 0 \cdot \mathbf{A} = \mathbf{0} \)
4. Real-World Example: Rain and Wind
- Rain falls vertically at 35 m/s (vr). Wind blows east-west at 12 m/s (vw).
- Resultant speed:
\( R = \sqrt{v_r^2 + v_w^2} = \sqrt{35^2 + 12^2} = 37 \, \text{m/s} \) - Direction:
\( \theta = \tan^{-1}\left(\frac{v_w}{v_r}\right) = \tan^{-1}(0.343) \approx 19^\circ \) from vertical. - Takeaway: The boy should tilt his umbrella 19° east from vertical.
5. Resolving Vectors
- Any vector A can be split into components along two non-parallel vectors a and b:
\( \mathbf{A} = \lambda \mathbf{a} + \mu \mathbf{b} \) - Example: Use parallelogram lines to find scaling factors λ and μ.
Important Concepts for NEET
- Resultant Vector Calculation: Use Pythagorean theorem and trigonometry (e.g., rain-wind problem).
- Vector Subtraction: Defined as adding the negative vector.
- Null Vector: Arises when a vector is subtracted from itself.
- Graphical Methods: Head-to-tail and parallelogram techniques for adding vectors.
- Direction Determination: Using \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \) (critical for physics problems).
