Alternative University

Analytic Geometry

Geometric Vectors

“Coordinates and vectors − in one form or another − are two of the most fundamental concepts in any discussion of mathematics as applied to physical problems.”
Sadri Hassani, Mathematical Methods for Students of Physics and Related Fields 2nd ed

Moving on to two dimensions, we can define vectors with two components to represent points on a rectangular grid that represents a plane:

Figure 1: Vectors with two components representing points on a rectangular grid that represents a plane.

These are referred to as geometric vectors. Each component of a geometric vector is an offset on a coordinate axis that is a number line. In this example, the x-axis and the y-axis are the coordinate axes.

Each geometric vector is essentially a directed line segment, with magnitude and direction.

A line segment is the shortest (straightest) distance between two points, those two points referred to as the end points of the line segment. The end point that marks the beginning of the vector is called the base or tail of the vector, and the end point at the other end (in the direction that the vector points) is called the head of the vector (usually denoted with an arrow head).

Figure 2: Position vectors.

Figure 2 shows the geometric vectors of Figure 1 as position vectors. Position vectors are fixed vectors: the base (tail) of a fixed vector is fixed at a point on the coordinate grid (in this example the origin), with the head at a point that is pointed to (in this example, point positions on the coordinate grid).

Besides fixed vectors, geometric vectors could be sliding vectors or free vectors. A sliding vector can slide anywhere along a line. That is useful in mechanics:

Figure 3: Moving a force along its line of action results in a new force system which is equivalent to the original force system. [UNL]

A free vector is a geometric vector that can be positioned anywhere, not just on a line, and may be referred to as parallel vectors, since each instance (positioning) of a free vector has the same direction (making the instances parallel):

Figure 4: Free vector representation of the strength and direction of gravity near the Earth surface. [MIT]

Scalar Multiplication

Scalar multiplication of a geometric vector scales the vector and reverses the direction if the scalar is negative.

In two dimensions, each of the two vector components are multiplied by the scalar:

λ( a, b )  =  ( λa, λb )

where λ is a scalar, and a and b are vector components.

For example, multiplying a geometric vector by −1.5 produces a vector that is 1.5 times as long and points in the opposite direction:

Figure 5: Vectors (2,3) and −1.5(2,3).

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2021–Sep–22  21:25  UTC