A **vector** is a mathematical object that has both magnitude (length) and direction. Vectors are used in various fields such as physics, engineering, and mathematics to represent quantities that have both size and direction, like force, velocity, or displacement.

### **Key Characteristics of Vectors:**
1. **Magnitude**: The length or size of the vector. It is a non-negative scalar quantity.
2. **Direction**: The orientation of the vector in space. It shows the direction in which the vector is acting.

### **Notation:**
- Vectors are usually denoted by boldface letters (e.g., **v**) or with an arrow above the letter (e.g., \(\vec{v}\)).
- A vector can be represented as \(\vec{v} = \begin{bmatrix} v_x \\ v_y \end{bmatrix}\) in 2D or \(\vec{v} = \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}\) in 3D, where \(v_x\), \(v_y\), and \(v_z\) are the components of the vector along the x, y, and z axes respectively.

### **Examples of Vectors:**

1. **Displacement Vector**:
   - Suppose you walk 5 meters to the east. This displacement can be represented as a vector \(\vec{d}\).
   - If you then walk 3 meters to the north, your total displacement can be represented by a vector \(\vec{d} = \begin{bmatrix} 5 \\ 3 \end{bmatrix}\) in 2D coordinates (where east is the positive x-direction and north is the positive y-direction).

2. **Velocity Vector**:
   - Imagine a car moving at a speed of 60 km/h toward the north-east direction. This can be represented by a velocity vector.
   - If the direction is at a 45-degree angle between the north and east, the vector might be written as \(\vec{v} = \begin{bmatrix} 60 \times \cos(45^\circ) \\ 60 \times \sin(45^\circ) \end{bmatrix}\).

3. **Force Vector**:
   - If a force of 10 Newtons acts on an object at an angle of 30 degrees from the horizontal, the force vector can be represented as:
   \[
   \vec{F} = \begin{bmatrix} 10 \times \cos(30^\circ) \\ 10 \times \sin(30^\circ) \end{bmatrix}
   \]

### **Operations with Vectors:**

1. **Vector Addition**:
   - If \(\vec{a} = \begin{bmatrix} a_x \\ a_y \end{bmatrix}\) and \(\vec{b} = \begin{bmatrix} b_x \\ b_y \end{bmatrix}\), their sum is \(\vec{a} + \vec{b} = \begin{bmatrix} a_x + b_x \\ a_y + b_y \end{bmatrix}\).
   
2. **Scalar Multiplication**:
   - Multiplying a vector by a scalar changes its magnitude but not its direction. If \(k\) is a scalar, then \(k\vec{v} = \begin{bmatrix} k v_x \\ k v_y \end{bmatrix}\).

3. **Dot Product**:
   - The dot product of two vectors \(\vec{a}\) and \(\vec{b}\) is a scalar, calculated as \(\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y\).

### **Visualization:**
- A vector is often represented graphically as an arrow. The length of the arrow corresponds to the magnitude, and the arrow points in the direction of the vector.

### **Applications of Vectors:**
- **Physics**: Representing forces, velocity, and acceleration.
- **Computer Graphics**: Modeling movement, lighting, and transformations.
- **Navigation**: Describing directions and distances.

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Would you like more details or have any questions?

### **Related Questions:**
1. How do you calculate the magnitude of a vector?
2. What is the difference between a vector and a scalar?
3. How are vectors used in physics to represent forces?
4. Can you explain vector cross product with an example?
5. What are unit vectors and how are they used?

**Tip:** Always ensure that when adding vectors, both vectors are expressed in the same coordinate system or basis.

Explore the concept of vectors with examples from physics and mathematics. Learn about vector representation, operations like addition and scalar multiplication, and applications in various fields. Ideal for higher education students and professionals.

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