What are vectors?
- Quantities that have both magnitude (size) and direction.
- Represented by arrows:
- Length of arrow = magnitude
- Direction of arrow = direction
Examples of vectors:
- Force
- Velocity
- Acceleration
- Displacement
Types of vectors:
- Free vectors: Can be moved without changing their effect.
- Sliding vectors: Can be moved along their line of action without changing their effect.
- Fixed vectors: Must be applied at a specific point.
Vector operations:
- Addition:
- Parallelogram law: Draw arrows representing the vectors head-to-tail. Complete the parallelogram. The diagonal from the starting point to the ending point represents the resultant vector.
- Triangle law: Similar to parallelogram law, but only draw two sides of the parallelogram.
- Subtraction:
- Add the negative of the vector to be subtracted.
- Multiplication by a scalar:
- Changes the magnitude of the vector, but not its direction.
- Dot product (scalar product):
- Produces a scalar quantity.
- Formula: A · B = |A| |B| cos θ (where θ is the angle between the vectors)
- Cross product (vector product):
- Produces a vector quantity perpendicular to the plane containing the two vectors.
- Formula: A × B = |A| |B| sin θ n (where n is a unit vector perpendicular to the plane)
Applications of vectors:
- Physics (force, velocity, acceleration)
- Engineering (structural analysis, fluid mechanics)
- Computer graphics (3D modeling, animation)
Additional notes:
- Vectors can be represented in component form (e.g., A = Ax i + Ay j + Az k).
- The angle between two vectors can be found using the dot product.
- The area of a parallelogram can be found using the cross product.
