Vectors: A Quick Reference
What are Vectors?
- Quantities that have both magnitude (size) and direction.
- Represented by arrows:
- The length of the arrow represents magnitude.
- The direction of the arrow represents direction.
Key Concepts
- Unit Vectors: Vectors of magnitude 1 used to indicate direction.
- i, j, k are the standard unit vectors in the x, y, and z directions, respectively.
- Vector Addition:
- Head-to-tail method: Place the tail of one vector at the head of the other.
- Parallelogram method: Form a parallelogram using the two vectors as adjacent sides. The diagonal from the common vertex represents the sum.
- Vector Subtraction:
- Add the negative of the vector to be subtracted.
- Scalar Multiplication:
- Multiplying a vector by a scalar changes its magnitude but not its direction.
- Dot Product:
- A scalar quantity obtained by multiplying the magnitudes of two vectors and the cosine of the angle between them.
- Used to find the angle between two vectors or the projection of one vector onto another.
- Cross Product:
- A vector quantity perpendicular to the plane containing the two vectors.
- Used to find the area of a parallelogram or the torque exerted by a force.
Applications
- Physics: Force, velocity, acceleration, momentum
- Engineering: Structural analysis, fluid dynamics
- Computer graphics: 3D modeling, animation
Examples
- Displacement: The change in position from a starting point to an ending point.
- Velocity: The rate of change of displacement.
- Force: A push or pull on an object.
- Momentum: The product of mass and velocity.