Vector Physics Quick Revision Sheets
What are vectors?
- Quantities that have both magnitude and direction.
- Examples: displacement, velocity, acceleration, force, momentum.
How are vectors represented?
- Arrow notation: The length of the arrow represents magnitude, and the direction of the arrow represents direction.
- Component form: Vectors can be broken down into their horizontal and vertical components.
Vector operations:
- Addition: Place the tail of one vector at the head of the other. The resultant vector is drawn from the tail of the first vector to the head of the second.
- Subtraction: Add the negative of the vector being subtracted.
- Scalar multiplication: Multiply the magnitude of the vector by the scalar. The direction remains the same if the scalar is positive, and reverses if the scalar is negative.
Vector components:
- The horizontal component of a vector can be found using:
Vx = V * cos(θ). - The vertical component of a vector can be found using:
Vy = V * sin(θ).
Vector magnitude and direction:
- The magnitude of a vector can be found using the Pythagorean theorem:
V = √(Vx² + Vy²). - The direction of a vector can be found using the tangent function:
θ = tan⁻¹(Vy / Vx).
Dot product:
- The dot product of two vectors A and B is defined as:
A · B = |A| * |B| * cos(θ), where θ is the angle between the vectors. - The dot product is used to find the projection of one vector onto another, the work done by a force, and the angle between two vectors.
Cross product:
- The cross product of two vectors A and B is defined as:
A × B = |A| * |B| * sin(θ) * n, where θ is the angle between the vectors, and n is a unit vector perpendicular to both A and B. - The cross product is used to find the torque exerted by a force, the angular momentum of a rotating object, and the area of a parallelogram.
Key formulas:
Vx = V * cos(θ)Vy = V * sin(θ)V = √(Vx² + Vy²)θ = tan⁻¹(Vy / Vx)A · B = |A| * |B| * cos(θ)A × B = |A| * |B| * sin(θ) * n
Remember to practice solving problems to reinforce your understanding of vector concepts.
