What Is A Direction Vector

Vectors and Direction

A report of motion will involve the introduction of a variety of quantities that are used to describe the physical world. Examples of such quantities include altitude, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. All these quantities can past divided into two categories - vectors and scalars. A vector quantity is a quantity that is fully described past both magnitude and direction. On the other manus, a scalar quantity is a quantity that is fully described past its magnitude. The emphasis of this unit of measurement is to sympathise some fundamentals about vectors and to use the fundamentals in order to understand motion and forces that occur in two dimensions.

Examples of vector quantities that have been previously discussed include displacement, velocity, acceleration, and force. Each of these quantities are unique in that a full description of the quantity demands that both a magnitude and a direction are listed. For instance, suppose your teacher tells yous "A handbag of gilded is located outside the classroom. To find it, displace yourself twenty meters." This statement may provide yourself enough information to pique your interest; yet, in that location is not enough information included in the argument to find the bag of gold. The displacement required to find the bag of gilt has not been fully described. On the other hand, suppose your teacher tells you "A purse of gilded is located outside the classroom. To detect information technology, readapt yourself from the center of the classroom door xx meters in a direction 30 degrees to the west of north." This argument now provides a complete description of the displacement vector - information technology lists both magnitude (twenty meters) and direction (30 degrees to the due west of due north) relative to a reference or starting position (the middle of the classroom door). Vector quantities are not fully described unless both magnitude and management are listed.

Representing Vectors

Vector quantities are often represented by scaled vector diagrams. Vector diagrams draw a vector past employ of an arrow drawn to scale in a specific management. Vector diagrams were introduced and used in earlier units to describe the forces acting upon an object. Such diagrams are commonly called as costless-torso diagrams. An case of a scaled vector diagram is shown in the diagram at the correct. The vector diagram depicts a displacement vector. Notice that there are several characteristics of this diagram that get in an appropriately fatigued vector diagram.

a scale is conspicuously listed
a vector arrow (with arrowhead) is drawn in a specified management. The vector arrow has a head and a tail.
the magnitude and management of the vector is clearly labeled. In this instance, the diagram shows the magnitude is 20 one thousand and the direction is (30 degrees Due west of North).

Conventions for Describing Directions of Vectors

Vectors tin be directed east, due West, due Southward, and due Due north. Merely some vectors are directed northeast (at a 45 degree angle); and some vectors are even directed northeast, even so more north than east. Thus, there is a clear need for some form of a convention for identifying the management of a vector that is not eastward, w, due South, or due Northward. In that location are a variety of conventions for describing the direction of whatever vector. The two conventions that volition be discussed and used in this unit of measurement are described below:

The direction of a vector is often expressed equally an angle of rotation of the vector about its "tail" from eastward, westward, n, or s. For example, a vector tin can be said to have a direction of twoscore degrees North of West (meaning a vector pointing West has been rotated 40 degrees towards the northerly direction) of 65 degrees East of South (meaning a vector pointing Due south has been rotated 65 degrees towards the easterly direction).
The direction of a vector is often expressed as a counterclockwise bending of rotation of the vector about its "tail" from east. Using this convention, a vector with a direction of 30 degrees is a vector that has been rotated xxx degrees in a counterclockwise direction relative to due e. A vector with a direction of 160 degrees is a vector that has been rotated 160 degrees in a counterclockwise direction relative to due due east. A vector with a management of 270 degrees is a vector that has been rotated 270 degrees in a counterclockwise management relative to due due east. This is one of the well-nigh common conventions for the direction of a vector and will be utilized throughout this unit of measurement.

Ii illustrations of the 2nd convention (discussed in a higher place) for identifying the direction of a vector are shown below.

Observe in the first example that the vector is said to have a direction of 40 degrees. You can remember of this management as follows: suppose a vector pointing Eastward had its tail pinned down and then the vector was rotated an bending of 40 degrees in the counterclockwise direction. Detect in the second example that the vector is said to have a direction of 240 degrees. This means that the tail of the vector was pinned down and the vector was rotated an angle of 240 degrees in the counterclockwise direction beginning from due east. A rotation of 240 degrees is equivalent to rotating the vector through 2 quadrants (180 degrees) and and so an additional threescore degrees into the 3rd quadrant.

Representing the Magnitude of a Vector

The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn a precise length in accordance with a chosen scale. For example, the diagram at the correct shows a vector with a magnitude of twenty miles. Since the scale used for constructing the diagram is 1 cm = 5 miles, the vector pointer is drawn with a length of iv cm. That is, four cm 10 (v miles/1 cm) = twenty miles.

Using the aforementioned scale (1 cm = 5 miles), a displacement vector that is 15 miles volition be represented by a vector pointer that is 3 cm in length. Similarly, a 25-mile deportation vector is represented by a 5-cm long vector arrow. And finally, an xviii-mile displacement vector is represented past a 3.6-cm long arrow. Run across the examples shown below.

In conclusion, vectors can be represented by use of a scaled vector diagram. On such a diagram, a vector arrow is drawn to represent the vector. The arrow has an obvious tail and arrowhead. The magnitude of a vector is represented by the length of the pointer. A scale is indicated (such every bit, 1 cm = v miles) and the arrow is drawn the proper length according to the chosen calibration. The arrow points in the precise direction. Directions are described by the use of some convention. The about common convention is that the direction of a vector is the counterclockwise angle of rotation which that vector makes with respect to due East.

In the remainder of this lesson, in the unabridged unit, and in hereafter units, scaled vector diagrams and the higher up convention for the direction of a vector will be often used to describe motion and solve problems apropos movement. For this reason, it is critical that you have a comfortable agreement of the ways of representing and describing vector quantities. Some exercise problems are bachelor on site at the following web page: