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Geometric & Algebraic Representations of Vectors

Instructor: Nadia Beherens

I have worked at several schools and universities as a Math teacher. I have a degree in Math teaching, and I am a master student.

In this lesson we will work the concept of vector and the difference between geometric and algebraic vector. We will develop definitions, properties and examples.

Vectors

In physics we can determine temperatures, time, forces, and displacement. Although they all represent quantities, there is a significant difference between them. Temperature and time are described only by magnitude, not by direction. In other words, if you measure the temperature of a place to be 25ºC, you cannot say that there is 25º C up or to the right. Temperature and time are measured in scalars.

However, you can say that you have advanced 5 meters in some direction (for example forward) or that you went 60 km/h from one corner to another. Forces and displacement are quantities described by a magnitude, but also by direction, and we call them vectors.

What is a Vector?

We can represent a vector as a line segment oriented from an initial point, called the tail, to a final point, called the head.

Each vector represents a numerical value. With vectors we can better explain problems that have to do with velocities, displacements, forces, and accelerations.

Types of Vectors

We can represent a vector as a line segment oriented from an initial point, called the tail, to a final point, called the head.

Geometric Vectors

Geometric vectors are not related to any coordinate system.

A geometric vector is not related to any coordinate system. A is the tail, B is the head.
AB image

Algebraic vectors

Algebraic vectors are related to a coordinate system.


(1,2) is the point located in the coordinate system, where 1 corresponds to x and 2 to y.
v12 image


Within this algebraic vectors, we can have a position vector that connects the origin of the coordinate system (O) with any point (P), and we write it like this:


OP


For example:


The vector connects the origin with the point A(2,3).
OA image


Algebraic vectors can also be a displacement vector, which is the vector from a point (A) to another one (B) and we write it like this:


AB


For example:


The vector connects the point A with the point B.
null


Vector Properties

All vectors have two fundamental properties; they have a magnitude and direction. The magnitude is the length, size, and norm of the vector and we can denote it by:


normv


To calculate it, we need to remember the Pythagorean Theorem: a2 + b2 = c2. So, if we have two points, A(x0 , y0 ) and B(x1 , y1 ), the magnitude of the vector AB will be:


magnitude


For example, if we have the vector AB with A(1,2) and B (3,5), the magnitude will be:


null


On the other hand, the direction is the measure of the angle it makes with a horizontal line. Then, if there are two vectors with the same magnitude and direction, we say that they are equivalent.


equivalent


In this case, all the vector have the same magnitude and direction. So, they are equivalent.


MNOP


While if they have the same magnitude and opposite direction, they are opposite.


OPPOSIT


So,


ab-ba


Transformation Example

What can we do to transform a displacement vector into a position vector? Let's answer this with an example:

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