Vectors

Vectors are a certain type of mathematical method that is often used in mechanical physics.

The main ideas include that scalars, such as temperature, have magnitude only. They are specified by a number with a unit and obey the rules of arithmetic and ordinary algebra. Vectors, such as displacement, have both magnitude and direction and obey the rules of vector algebra.

Vector Notation
Two vectors 'a-arrow' and 'b-arrow' may be added geometrically by drawing them to a common scale and placing them head to tail. The vector connecting the tail of the first to the head of the second is the vector sum (often denoted as 's-arrow'). To subtract 'b-arrow' from 'a-arrow', reverse the direction of 'b-arrow' to get the negative 'b-arrow', then add the negative 'b-arrow' to 'a-arrow'. Vector addition is commutative and obeys the associative law.

The scalar components a(x) and a(y) of any two-dimensional vector 'a-arrow' along the coordinate axes are found by dropping perpendicular lines from the ends of 'a-arrow' onto the coordinate axes. The components are given by the two equations (a(x) = acos(theta)) and (a(y) = asin(theta)). These equations are compatible where the angle (theta) between the positive direction of the x axis and the direction of 'a-arrow'. The algebraic sign of a component indicates its direction along the associated axis. Given its components, we can find the magnitude and orientation of the vector 'a-arrow' with the Pythagoras equations.