Summary
Highlights
Vectors are quantities with both magnitude and direction. A 2D vector 'v' has components Vx (V cos θ) and Vy (V sin θ). Magnitude is denoted by |v| and is calculated using the Pythagorean theorem for 2D and 3D vectors. The angle θ, found with arctan(Vy/Vx), determines the vector's direction.
A vector can be written in terms of its components using standard unit vectors i, j, and k representing the x, y, and z directions, respectively. A unit vector has a magnitude of one and its purpose is to specify direction.
The dot product of two vectors A and B results in a scalar quantity. It's calculated as |A||B| cos θ, or by summing the products of their corresponding components (AxBx + AyBy + AzBz). An example application is calculating work, where Work = Force ⋅ Displacement.
The cross product of two vectors A and B yields another vector that is perpendicular to both. It's calculated as |A||B| sin θ multiplied by a unit vector representing the new direction. The vector components of the cross product can also be found by evaluating a 3x3 determinant.
The cross product is used to calculate torque (τ = r × F), where r is the lever arm and F is the force. It's also fundamental in determining the magnetic force on a moving charge (F_magnetic = Q (v × B)), where Q is the charge, v is velocity, and B is the magnetic field.
Any vector can be expressed as its magnitude multiplied by its unit vector (v = |v| û). A unit vector (û) is found by dividing a vector by its magnitude. The position vector describes the location of a point relative to the origin, and its unit vector specifies the direction of that position.
The electric field vector (E) at a point due to a charge Q is given by E = (kQ/R²) R̂, where k is Coulomb's constant, R is the distance, and R̂ is the unit vector of the position vector. This formula shows how magnitude and direction combine to define the electric field.