Vector Addition and Subtraction: Graphical Methods Quiz

PDF page 125: A vector is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector’s magnitude and pointing in the direction of the vector.

PDF page 125: A vector is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector’s magnitude and pointing in the direction of the vector.

PDF page 125: A vector is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector’s magnitude and pointing in the direction of the vector.

PDF page 126: FIGURE 3.10 Head-to-Tail Method: The head-to-tail method of graphically adding vectors is illustrated for the two displacements of the

PDF page 127: 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.

PDF page 127: 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.

PDF page 127: Step 5. To get the magnitude of the resultant, measure its length with a ruler. (Note that in most calculations, we

PDF page 128: 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.

PDF page 128: 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.

PDF page 129: 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

PDF page 129: In this case, the total displacement is seen to have a magnitude of 50.8 m and to lie in a direction south of east. By using its magnitude and direction, this vector can be expressed as and 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 Figure 3.18 and we will still get the same solution.

PDF page 130: Vector subtraction is a straightforward extension of vector addition. To define subtraction (say we want to subtract from , written , we must first define what we mean by subtraction. The negative of a vector is defined to be ; that is, graphically the negative of any vector has the same magnitude but the opposite direction, as shown in

PDF page 130: FIGURE 3.19 The negative of a vector is just another vector of the same magnitude but pointing in the opposite direction. So is the

PDF page 130: to . Note that vector subtraction is the addition of a negative vector. The order of subtraction does not affect the results. 3.2 This is analogous to the subtraction of scalars (where, for example, ). Again, the result is independent of the order in which the subtraction is made. When vectors are subtracted graphically, the techniques outlined above are used, as the following example illustrates.