Question

Addition and subtraction of vectors

Answer 1

Vectors are quantities that have both magnitude and direction. In mathematics, vectors can be added or subtracted using geometric or algebraic methods. Below are details on how to perform vector addition and subtraction.

### Vector Addition

1. **Geometric Method:**
- The head-to-tail method: To add two vectors, place the tail of the second vector at the head of the first vector. The resultant vector is drawn from the tail of the first vector to the head of the second vector.
- Parallelogram method: If two vectors are placed with their tails at the same point, you can complete a parallelogram where the two vectors are adjacent sides. The diagonal of the parallelogram represents the resultant vector.

2. **Algebraic Method:**
- If vectors **A** and **B** are represented in components:
\[
\mathbf{A} = (A_x, A_y) \quad \text{and} \quad \mathbf{B} = (B_x, B_y)
\]
The resultant vector **R** is given by:
\[
\mathbf{R} = \mathbf{A} + \mathbf{B} = (A_x + B_x, A_y + B_y)
\]

### Vector Subtraction

1. **Geometric Method:**
- To subtract vector **B** from vector **A** (i.e., **A** - **B**), you can add vector **A** to the negative of vector **B**. This means you reverse the direction of vector **B** and then use the head-to-tail method or parallelogram method for addition.

2. **Algebraic Method:**
- Using components, to subtract vector **B** from vector **A**:
\[
\mathbf{R} = \mathbf{A} - \mathbf{B} = (A_x - B_x, A_y - B_y)
\]

### Example

Let's consider the vectors:
\[
\mathbf{A} = (3, 4) \quad \text{and} \quad \mathbf{B} = (1, 2)
\]

**Addition:**
\[
\mathbf{R} = \mathbf{A} + \mathbf{B} = (3 + 1, 4 + 2) = (4, 6)
\]

**Subtraction:**
\[
\mathbf{R} = \mathbf{A} - \mathbf{B} = (3 - 1, 4 - 2) = (2, 2)
\]

### Summary

- **Addition** of vectors combines their components.
- **Subtraction** of vectors involves subtracting their components.
- Vectors can be represented graphically or algebraically, and both methods will provide the same result when performed correctly.