Vectors introduction pdf

An example of the use of the head-to-tail method is illustrated below. The problem involves the addition of three vectors:. The head-to-tail method is employed as described above and the resultant is determined drawn in red. Its magnitude and direction is labeled on the diagram. Interestingly enough, the order in which three vectors are added has no affect upon either the magnitude or the direction of the resultant.

The resultant will still have the same magnitude and direction. For example, consider the addition of the same three vectors in a different order. When added together in this different order, these same three vectors still produce a resultant with the same magnitude and direction as before The order in which vectors are added using the head-to-tail method is insignificant.

It is the result of adding two or more vectors together. If displacement vectors A, B, and C are added together, the result will be vector R. As shown in the diagram, vector R can be determined by the use of an accurately drawn, scaled, vector addition diagram. To say that vector R is the resultant displacement of displacement vectors A, B, and C is to say that a person who walked with displacements A, then B, and then C would be displaced by the same amount as a person who walked with displacement R.

That is why it can be said that. When displacement vectors are added, the result is a resultant displacement. But any two vectors can be added as long as they are the same vector quantity. If two or more velocity vectors are added, then the result is a resultant velocity. If two or more force vectors are added, then the result is a resultant force. If two or more momentum vectors are added, then the result is In all such cases, the resultant vector whether a displacement vector, force vector, velocity vector, etc.

The football player experiences three different applied forces. Each applied force contributes to a total or resulting force. If the three forces are added together using methods of vector addition , then the resultant vector R can be determined. In this case, to experience the three forces A, B and C is the same as experiencing force R. To be hit by players A, B, and C would result in the same force as being hit by one player applying force R.

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Search inside document. The two conventions that will be discussed and used in this unit are described below: a. To see how the method works, consider the following problem: Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. The problem involves the addition of three vectors: 20 m, 45 deg.

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Prince Yug. Mohammad Ali. Anubhab Dutta Gupta. Amir Akmal. More From Nathan Mwansa. Both of these properties must be given in order to specify a vector completely. A b a 1 b 1 diagram 2 in diagram 2 the vectors ab and a 1b 1 are equal i e. Introduction to name vectors and angles physicsfundamentals gpb 2 04 1. In this unit we describe how to write down vectors how to add and subtract them and how to use them in geometry.

Vectors introduction 4 two vectors are equal if they have the same magnitude the same direction i e. If two vectors have the same length are parallel but have opposite senses then one is the. Intro To Vectors Worksheet. Pin On Math. Your email address will not be published. For example, a vector can be said to have a direction of 40 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 South has been rotated 65 degrees towards the easterly direction.

The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its "tail" from due East. Using this convention, a vector with a direction of 30 degrees is a vector that has been rotated 30 degrees in a counterclockwise direction relative to due east. A vector with a direction of degrees is a vector that has been rotated degrees in a counterclockwise direction relative to due east.

This is one of the most common conventions for the direction of a vector and will be utilized throughout this unit. Two illustrations of the second convention discussed above 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 think of this direction as follows: suppose a vector pointing East had its tail pinned down and then the vector was rotated an angle of 40 degrees in the counterclockwise direction.

Observe in the second example that the vector is said to have a direction of degrees. This means that the tail of the vector was pinned down and the vector was rotated an angle of degrees in the counterclockwise direction beginning from due east. A rotation of degrees is equivalent to rotating the vector through two quadrants degrees and then an additional 60 degrees into the third 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 right shows a vector with a magnitude of 20 miles. Similarly, a mile displacement vector is represented by a 5-cm long vector arrow.

And finally, an mile displacement vector is represented by a 3. See 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 arrow.

The arrow points in the precise direction. Directions are described by the use of some convention. The most common convention is that the direction of a vector is the counterclockwise angle of rotation which that vector makes with respect to due East.

Vector Addition A variety of mathematical operations can be performed with and upon vectors. One such operation is the addition of vectors. Two vectors can be added together to determine the result or resultant.

Sample applications are shown in the diagram below. In this unit, the task of summing vectors will be extended to more complicated cases in which the vectors are directed in directions other than purely vertical and horizontal directions. For example, a vector directed up and to the right will be added to a vector directed up and to the left. The vector sum will be determined for the more complicated cases shown in the diagrams below.

The Pythagorean Theorem The Pythagorean theorem is a useful method for determining the result of adding two and only two vectors that make a right angle to each other.

The method is not applicable for adding more than two vectors or for adding vectors that are not at degrees to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle. Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement. This problem asks to determine the result of adding two displacement vectors that are at right angles to each other.

The result or resultant of walking 11 km north and 11 km east is a vector directed northeast as shown in the diagram to the right. Since the northward displacement and the eastward displacement are at right angles to each other, the Pythagorean theorem can be used to determine the resultant i.

The result of adding 11 km, north plus 11 km, east is a vector with a magnitude of Using Trigonometry to Determine a Vector's Direction The direction of a resultant vector can often be determined by use of trigonometric functions. These three functions relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle. The sine function relates the measure of an acute angle to the ratio of the length of the side opposite the angle to the length of the hypotenuse.

The cosine function relates the measure of an acute angle to the ratio of the length of the side adjacent the angle to the length of the hypotenuse. The tangent function relates the measure of an angle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

The three equations below summarize these three functions in equation form. These three trigonometric functions can be applied to the hiker problem in order to determine the direction of the hiker's overall displacement. The process begins by the selection of one of the two angles other than the right angle of the triangle.

Once the angle is selected, any of the three functions can be used to find the measure of the angle. Write the function and proceed with the proper algebraic steps to solve for the measure of the angle. The work is shown below. Once the measure of the angle is determined, the direction of the vector can be found. In this case the vector makes an angle of 45 degrees with due East. Thus, the direction of this vector is written as 45 degrees. The following vector addition diagram is an example of such a situation.

Observe that the angle within the triangle is determined to be This angle is the southward angle of rotation that the vector R makes with respect to West. Yet the direction of the vector as expressed with the CCW counterclockwise from East convention is Use of Scaled Vector Diagrams to Determine a Resultant The magnitude and direction of the sum of two or more vectors can also be determined by use of an accurately drawn scaled vector diagram.

Using a scaled diagram, the head-to-tail method is employed to determine the vector sum or resultant. A common Physics lab involves a vector walk. Either using centimeter-sized displacements upon a map or meter-sized displacements in a large open area, a student makes several consecutive displacements beginning from a designated starting position. Suppose that you were given a map of your local area and a set of 18 directions to follow.

Starting at home base, these 18 displacement vectors could be added together in consecutive fashion to determine the result of adding the set of 18 directions. Perhaps the first vector is measured 5 cm, East.

Where this measurement ended, the next measurement would begin. The process would be repeated for all 18 directions. Each time one measurement ended, the next measurement would begin. In essence, you would be using the head-to-tail method of vector addition. The head-to-tail method involves drawing a vector to scale on a sheet of paper beginning at a designated starting position.