A **vector** is a mathematical object that has both a magnitude (size) and a direction. Vectors are used to represent quantities that involve both of these aspects, such as displacement, velocity, force, and acceleration.

### Key Features of Vectors:
1. **Magnitude**: The length or size of the vector. It is often represented by the absolute value or the norm of the vector.
2. **Direction**: The way the vector is pointing, usually described by angles or in terms of its components in a coordinate system.

### Representation of Vectors:
- **Graphically**: A vector is often represented as an arrow. The length of the arrow represents the magnitude, and the direction of the arrow represents the direction of the vector.
- **Component form**: In 2D or 3D space, vectors are expressed in terms of their components along the x, y (and possibly z) axes. For example:
  - In 2D: \( \mathbf{v} = \langle v_x, v_y \rangle \)
  - In 3D: \( \mathbf{v} = \langle v_x, v_y, v_z \rangle \)
- **Unit vectors**: These are vectors of length 1, used to indicate direction. For example, in 2D, \( \hat{i} \) and \( \hat{j} \) are unit vectors along the x and y axes, respectively.

### Vector Operations:
1. **Addition**: Vectors can be added together by adding their corresponding components.
   - \( \mathbf{v} + \mathbf{w} = \langle v_x + w_x, v_y + w_y \rangle \)
2. **Scalar Multiplication**: A vector can be multiplied by a scalar (a number), which scales its magnitude.
   - \( c \mathbf{v} = \langle c v_x, c v_y \rangle \)
3. **Dot Product**: A scalar product of two vectors, which measures the cosine of the angle between them.
   - \( \mathbf{v} \cdot \mathbf{w} = v_x w_x + v_y w_y \)
4. **Cross Product**: A vector product (only in 3D), which results in a new vector perpendicular to both.
   - \( \mathbf{v} \times \mathbf{w} = \langle v_y w_z - v_z w_y, v_z w_x - v_x w_z, v_x w_y - v_y w_x \rangle \)

### Applications:
- **Physics**: Representing forces, velocity, acceleration.
- **Computer Graphics**: Defining direction and scaling of movements.
- **Engineering**: Used to model systems, such as stress and strain in materials.

Would you like more details or have any questions?

Here are five related questions to expand this concept:
1. How do you calculate the magnitude of a vector in 3D?
2. What is the geometric interpretation of the dot product?
3. How does vector addition differ from scalar addition?
4. Can vectors be subtracted? If so, how is this done?
5. What are some real-world examples where vectors are used?

**Tip**: When performing vector operations, always ensure to handle each component (x, y, z) correctly.

Learn about vectors, their key features, and how to perform vector operations like addition, scalar multiplication, dot product, and cross product. This comprehensive guide covers vector concepts used in physics, computer graphics, and engineering. Suitable for high school students in Grades 10-12.

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