<h2>Vector Addition: Head-to-Tail Method</h2>
The head-to-tail method is a graphical way to add vectors, described in the figure below below and in the steps following. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the final, pointed end of the arrow.
<div class="wp-caption alignnone" style="width: 800px"><img alt="In part a, a vector of magnitude of nine units and making an angle of theta is equal to zero degrees is drawn from the origin and along the positive direction of x axis. In part b a vector of magnitude of nine units and making an angle of theta is equal to zero degree is drawn from the origin and along the positive direction of x axis. Then a vertical arrow from the head of the horizontal arrow is drawn. In part c a vector D of magnitude ten point three is drawn from the tail of the horizontal vector at an angle theta is equal to twenty nine point one degrees from the positive direction of x axis. The head of the vector D meets the head of the vertical vector. A scale is shown parallel to the vector D to measure its length. Also a protractor is shown to measure the inclination of the vectorD." height="326" src="https://data.ulearngo.com/assets/03e5fb46-07ee-4aa1-a91f-95289c995c89" width="800"/>Head-to-Tail Method: The head-to-tail method of graphically adding vectors is illustrated for the two displacements of the person walking in a city considered in the previous lesson. (a) Draw a vector representing the displacement to the east. (b) Draw a vector representing the displacement to the north. The tail of this vector should originate from the head of the first, east-pointing vector. (c) Draw a line from the tail of the east-pointing vector to the head of the north-pointing vector to form the sum or resultant vector \(\mathbf{D}\). The length of the arrow \(\mathbf{D}\) is proportional to the vector’s magnitude and is measured to be 10.3 units . Its direction, described as the angle with respect to the east (or horizontal axis) \(\theta \) is measured with a protractor to be \(\text{29}\text{.}1º\).</div>
Step 1. Draw an arrow to represent the first vector (9 blocks to the east) using a ruler and protractor.
<img alt="In part a, a vector of magnitude of nine units and making an angle theta is equal to zero degree is drawn from the origin and along the positive direction of x axis." src="https://data.ulearngo.com/assets/5f81681a-6bab-4cb7-8cc5-65f65436bbd2"/>
Step 2. Now draw an arrow to represent the second vector (5 blocks to the north). Place the tail of the second vector at the head of the first vector.
<img alt="In part b, a vector of magnitude of nine units and making an angle theta is equal to zero degree is drawn from the origin and along the positive direction of x axis. Then a vertical vector from the head of the horizontal vector is drawn." src="https://data.ulearngo.com/assets/c3798a5b-e16b-4322-8eb8-9337391ef419"/>
Step 3. If there are more than two vectors, continue this process for each vector to be added. Note that in our example, we have only two vectors, so we have finished placing arrows tip to tail.
Step 4. Draw an arrow from the tail of the first vector to the head of the last vector. This is the resultant, or the sum, of the other vectors.
<img alt="In part c, a vector D of magnitude ten point three is drawn from the tail of the horizontal vector at an angle theta is equal to twenty nine point one degrees from the positive direction of the x axis. The head of the vector D meets the head of the vertical vector. A scale is shown parallel to the vector D to measure its length. Also a protractor is shown to measure the inclination of the vector D." src="https://data.ulearngo.com/assets/90fb5754-3941-4cdd-9c17-34fe97e7ecb9"/>
Step 5. To get the magnitude of the resultant, measure its length with a ruler. (Note that in most calculations, we will use the Pythagorean theorem to determine this length.)
Step 6. To get the direction of the resultant, measure the angle it makes with the reference frame using a protractor. (Note that in most calculations, we will use trigonometric relationships to determine this angle.)
The graphical addition of vectors is limited in accuracy only by the precision with which the drawings can be made and the precision of the measuring tools. It is valid for any number of vectors.
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<h3>Example: Adding Vectors Graphically Using the Head-to-Tail Method: A Woman Takes a Walk</h3>
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Use the graphical technique for adding vectors to find the total displacement of a person who walks the following three paths (displacements) on a flat field. First, she walks 25.0 m in a direction \(\text{49.0º}\) north of east. Then, she walks 23.0 m heading \(\text{15.0º}\) north of east. Finally, she turns and walks 32.0 m in a direction 68.0° south of east.
Strategy
Represent each displacement vector graphically with an arrow, labeling the first \(\mathbf{A}\), the second \(\mathbf{B}\), and the third \(\mathbf{C}\), making the lengths proportional to the distance and the directions as specified relative to an east-west line. The head-to-tail method outlined above will give a way to determine the magnitude and direction of the resultant displacement, denoted \(\mathbf{\mathbf{R}}\).
Solution
(1) Draw the three displacement vectors.
<img alt="On the graph a vector of magnitude twenty three meters and inclined above the x axis at an angle theta-b equal to fifteen degrees is shown. This vector is labeled as B." src="https://data.ulearngo.com/assets/0f961c2a-63d2-42a0-b21c-477ce5db0df2"/>
(2) Place the vectors head to tail retaining both their initial magnitude and direction.
<img alt="In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis." src="https://data.ulearngo.com/assets/e298a886-a6b5-4c3e-b04f-cb34c811090a"/>
(3) Draw the resultant vector, \(\mathbf{R}\).
<img alt="In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis. From the tail of the vector A a vector R of magnitude of fifty point zero meters and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector C. The vector R is known as the resultant vector." src="https://data.ulearngo.com/assets/3b979e44-5f77-4b1b-b181-70ea0da8798f"/>
(4) Use a ruler to measure the magnitude of \(\mathbf{\mathbf{R}}\), and a protractor to measure the direction of \(\mathbf{R}\). While the direction of the vector can be specified in many ways, the easiest way is to measure the angle between the vector and the nearest horizontal or vertical axis. Since the resultant vector is south of the eastward pointing axis, we flip the protractor upside down and measure the angle between the eastward axis and the vector.
<img alt="In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis. From the tail of the vector A a vector R of magnitude of fifty meter and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector C. The vector R is known as the resultant vector. A ruler is placed along the vector R to measure it. Also there is a protractor to measure the angle." src="https://data.ulearngo.com/assets/51886264-8be5-4ae0-80db-27f49ebd1450"/>
In this case, the total displacement \(\mathbf{\text{R}}\) is seen to have a magnitude of 50.0 m and to lie in a direction \(7.0º\) south of east. By using its magnitude and direction, this vector can be expressed as \(R=\text{50.0 m}\) and \(\theta =7\text{.}\text{0º}\) south of east.
Discussion
The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that the resultant is independent of the order in which the vectors are added. Therefore, we could add the vectors in any order as illustrated in the figure below and we will still get the same solution.
<img alt="In this figure a vector C with a negative slope is drawn from the origin. Then from the head of the vector C another vector A with positive slope is drawn and then another vector B with negative slope from the head of the vector A is drawn. From the tail of the vector C a vector R of magnitude of fifty point zero meters and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector B. The vector R is known as the resultant vector." src="https://data.ulearngo.com/assets/8fe4c257-79dd-4fcb-91f3-aa0e1303feb0"/>
Here, we see that when the same vectors are added in a different order, the result is the same. This characteristic is true in every case and is an important characteristic of vectors. Vector addition is commutative. Vectors can be added in any order.
\(\mathbf{\text{A}}+\mathbf{\text{B}}=\mathbf{\text{B}}+\mathbf{\text{A}}\text{.}\)
(This is true for the addition of ordinary numbers as well—you get the same result whether you add \(\mathbf{\text{2}}+\mathbf{\text{3}}\) or \(\mathbf{\text{3}}+\mathbf{\text{2}}\), for example).
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Vector Addition: Head-to-Tail Method

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Explore the principles of two-dimensional kinematics, including mastering vector addition and subtraction, resolving a vector into components, and analyzing projectile motion and relative velocities.


Vector Addition: Head-to-Tail Method

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