# Discovery Science: Analytic geometry – Vectors

## Analytic Geometry – Vectors

Vectors are used to describe movements in planes and space and are represented in a coordinate system by their components in x, y, and sometimes z directions.

A vector is often drawn as an arrow rep-resenting the movement of a point in space. The length (magnitude) of an arrow and its direction, make up a vector quantity. In geometry, all arrows with the same direction and length represent the same vector. A vector can be placed anywhere on a coordinate system and can also be defined by a pair or triplet of numbers, with the understanding that the arrow begins at the origin.

A vector space is a set of vectors and various specific operations in the form of an algebraic equation. Because the structural content is independent of real quantities, vector calculations are useful in analytic geometry, physics, and engineering.

Points and point sets

Vectors describe many different quantities. A velocity vector defines the direction and speed of an object. A position vector defines the location of a point relative to the origin of a coordinate system. The linear equation describing the set of points on a straight line is; x(->)= a(->) + rb(->). To reach x(->), start at a (the position vector) and travel a distance equal to r times b(->), in the direction specified by b (the velocity vector).

A plane can be represented by: x(->)= a(->) +rb(->) + sc(->) A second velocity vector sc(->) represents a distance equal to s times c, in the direction specified by c, on a different plane.

Geometry without drawings

Descriptions of geometrical objects can be depicted using calculations instead of diagrams. Using calculations, common points of different sets of points and distance calculations can be found.

In physics and computer graphics, complex problems in which vectors play a role are solved through pure calculation. Full-length animated movies also use vector graphics.

### COMPLEX NUMBERS AS VECTORS

Complex numbers expand the domain of real numbers by introducing a new value, 1, defined as the square root of -1, so that 1^2 = -1. Using this new number, the roots of negative as well as positive numbers can be calculated.

Complex numbers are represented according to the formula a + bi, whereby a and b are real numbers and 1 is an imaginary unit. One convenient way to represent complex numbers is the complex plane (Argand diagram: a coordinate system in which the x-axis represents the real component, a. and the y-axis the imaginary value, b).

Therefore every point on the plane corresponds to a unique complex number, making it comparable to a two-dimensional linear space or vector field. In accordance with their definition, the addition of complex numbers is comparable to that of vectors. However, their multiplication is I different; multiplying B two complex numbers produces a third vector of different magnitude and direction.

### VECTORS IN PHYSICS

Variables with both a magnitude and a direction are known as vectors. They include magnetic and electrical field-strength speed, acceleration, and force. However, mass and
energy are non directional and thus are considered scalar.

A communication satellite in orbit around the Earth at a constant speed would keep moving in a straight line due to its inertia; however, the acceleration of gravity applies a force perpendicular to the direction of its movement, which affects its motion; thus only the direction of the speed vector changes.