A vector is an amount or phenomenon that has two independent properties: magnitude and direction. The term also denotes the mathematical or geometric representation of such quantity.
Examples of vectors in nature are speed, force, electromagnetic fields and weight. An amount or phenomenon that only shows magnitude, without specific direction, is called scalar.
Examples of scalars include speed, mass, electrical resistance, and hard disk storage capacity.
The vectors can be represented graphically in two or three dimensions. The magnitude is shown as the length of a segment. The direction is shown by the orientation of the segment and by an arrow at one end.
The illustration above shows three vectors in two-dimensional rectangular coordinates (the Cartesian plane) and their equivalents in polar coordinates.
The vectors in physics
In physics, when you have a vector, you have to take into account two quantities: its direction and its magnitude. Quantities that are only one magnitude are called scalar. If a scalar magnitude is given an address, a vector is created.
Visually, you see vectors drawn as arrows, which is perfect because an arrow has a clear direction and a clear magnitude (the length of the arrow).
In the following figure the arrow represents a vector that starts at the foot of the arrow (also called the tail) and ends at the head.
In physics, a bold letter is usually used to represent a vector, although it can also be represented as a letter with an arrow above it.
The arrow means that it is not just a scalar value, which would be represented by A, but also something with direction.
Differences between vector and scalar
Values that are not vectors are scalar. For example, such an amount of 500 apples is a scalar, has no direction, is only a magnitude. Time is a scalar too, it has no direction.
However, velocity is a vector since it not only specifies a magnitude (velocity) of the path, it also indicates the direction (and direction) of the path.
For example, the line of action of the velocity vector may
Be 30 ° from the horizontal. Therefore, we know in which direction the object moves.
However, this still does not specify the meaning of the trip, whether it is moving away or approaching us. Therefore, we also specify the direction in which the vector acts through an arrowhead.
Force, acceleration and distance traveled are also vectors. For example, saying that a car moved 10 meters does not indicate in which direction it moved. To fully specify movement, you must also specify the direction and direction of movement.
Force is also a vector because if you pull an object towards you it comes close to you, and if you push the object it will move away from you. So the force has a direction and a sense, and therefore, it is a vector.
Example
As an example of the information provided by a vector we have the following:
Suppose a teacher tells you:"A bag of gold is outside the classroom, to find it, go 20 meters." This sure statement will arouse your interest, however, there is not enough information included in the statement to find the gold bag.
The displacement required to find the gold bag has not been fully described. On the other hand, suppose your teacher tells you,"A bag of gold is located outside the classroom, to find it, move from the center of the door of the class 20 meters in a direction 30 ° west of the north."
This statement now provides a complete description of the displacement vector, which lists the magnitude (20 meters) and the direction (30 ° west of the north) with respect to a reference or departure position (the center of the class gate ).
Vector quantities are not fully described unless both magnitude and direction are indicated.
Vector representation
Vector quantities are often represented by scaled vector diagrams.
The vector diagrams represent a vector by using an arrow drawn to scale in a specific direction. An appropriate vector diagram must have several characteristics:
- A scale is clearly listed.
- Draw a vector arrow (with arrowhead) in a specific direction. The vector arrow has a head and a tail.
- The magnitude and direction of the vector is clearly labeled.
Direction of a vector
Vectors can be directed east, west, south and north. But some vectors are directed to the northeast (at a 45 ° angle). Therefore, there is a clear need to identify the direction of a vector that does not depend on the north, south, east or west.
There are a variety of conventions to describe the direction of any vector, however, only two of them will be explained below.
1-The direction of a vector is often expressed as an angle of rotation of the vector around its"tail"towards the east, west, north or south.
For example, a vector may be said to have a direction of 40 ° north of the west (meaning that a vector pointing west has been rotated 40 ° northward) or has a direction of 65 ° degrees To the east of the south (meaning that a vector pointing south has rotated 65 ° eastward).
2-The direction of a vector is often expressed as an anti-clockwise rotation angle of the vector. Using this convention, a vector with a direction of 30 ° is a vector that has been rotated 30 ° in a counter-clockwise direction with respect to the east.
A vector with a direction of 160 ° is a vector that has been rotated 160 ° in a counterclockwise direction with respect to the east. A vector with a direction of 270 ° is a vector that has been rotated 270 ° in a counterclockwise direction relative to the east.
Magnitude of a vector
The magnitude of a vector in a scaled vector diagram is represented by the length of the arrow. The arrow is drawn to a precise length according to a chosen scale.
For example, if you want to draw a vector that has a magnitude of 20 meters, you can choose as scale 1 cm = 5 meters, and you would draw an arrow with a length of 4 cm.
Using the same scale (1 cm = 5 meters), a vector of displacement of 15 meters will be represented by a vector arrow of 3 cm in length.
Similarly, a 25-meter displacement vector is represented by an arrow 5 cm in length. And finally, a vector of displacement of 18 meters is represented by an arrow of 3.6 cm in length.
Other characteristics of vectors
Equality : Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal.
Opposition : Two vectors are opposite if they have the same magnitude but opposite direction.
Parallels : Two vectors are parallel if they have the same direction but not necessarily the same magnitude, or antiparallel if they have opposite direction but not necessarily the same magnitude.
Unit Vector : A unit vector is any vector with a length of one.
Vector zero : The vector zero is the vector with zero length. Unlike any other vector, it has an arbitrary or indeterminate direction, and can not be normalized
References
- Jong IC, Rogers BG. Engineering mechanics: statics (1991). Saunders College Publishing.
- Ito K. Encyclopedic Dictionary of Mathematics (1993). MIT Press.
- Ivanov AB. Encyclopedia of Mathematics (2001). Springer.
- Kane T, Levinson D. Dynamics Online (1996). Sunnyvale: OnLine Dynamics.
- Lang S. Introduction to linear algebra (1986). Springer.
- Niku S. Engineering principles in everyday life for non-engineers (2016). Morgan & Claypool.
- Pedoe D. Geometry: a comprehensive course (1988). Dover.