A vector is a mathematical object that has both magnitude and direction. It is commonly represented by an arrow pointing in the direction of the vector and its length represents its magnitude.
In vector analysis, $\varphi$ is the angle between two vectors. It is typically measured in radians or degrees and can be calculated using the dot product or trigonometric functions.
Vector addition is performed by placing the initial point of the second vector at the terminal point of the first vector. The resulting vector is the diagonal of the parallelogram formed by the two original vectors.
The cross product of two vectors is a vector that is perpendicular to both of the original vectors. Its magnitude is equal to the product of the magnitudes of the two vectors and the sine of the angle between them. Its direction is given by the right-hand rule.
Vector analysis is commonly used in physics to describe and analyze the motion and forces acting on objects. It is also used in electromagnetics to describe the direction and magnitude of electric and magnetic fields.