Vector Addition and Subtraction: Analytical Methods Quiz

PDF page 133: Analytical methods of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. However, analytical methods are more concise, accurate, and precise than graphical methods, which are limited by the accuracy with which a drawing can be made. Analytical methods are limited only by the accuracy and precision with which physical quantities are known.

PDF page 133: Analytical techniques and right triangles go hand-in-hand in physics because (among other things) motions along perpendicular directions are independent. We very often need to separate a vector into perpendicular components. For example, given a vector like in Figure 3.24, we may wish to find which two perpendicular vectors, and , add to produce it.

PDF page 133: and . These vectors form a right triangle. The analytical relationships among these vectors are summarized below. and

PDF page 133: are defined to be the components of along the x- and y-axes. The three vectors , , and

PDF page 133: form a right triangle: 3.3 Note that this relationship between vector components and the resultant vector holds only for vector quantities (which include both magnitude and direction). The relationship does not apply for the magnitudes alone. For example, if east, north, and north-east, then it is true that the vectors . However, it is not true that the sum of the magnitudes of the vectors is also equal. That is,

PDF page 133: are defined to be the components of along the x- and y-axes. The three vectors , , and

PDF page 133: form a right triangle: 3.3 Note that this relationship between vector components and the resultant vector holds only for vector quantities (which include both magnitude and direction). The relationship does not apply for the magnitudes alone. For example, if east, north, and north-east, then it is true that the vectors . However, it is not true that the sum of the magnitudes of the vectors is also equal. That is,

PDF page 135: Note that the equation is just the Pythagorean theorem relating the legs of a right triangle to the length of the hypotenuse. For example, if and are 9 and 5 blocks, respectively, then blocks, again consistent with the example of the person walking in a city. Finally, the direction is , as before.

PDF page 137: Step 3. To get the magnitude of the resultant, use the Pythagorean theorem:

PDF page 135: Note that the equation is just the Pythagorean theorem relating the legs of a right triangle to the length of the hypotenuse. For example, if and are 9 and 5 blocks, respectively, then blocks, again consistent with the example of the person walking in a city. Finally, the direction is , as before.

PDF page 136: and add to give the magnitude of the resultant vector in the horizontal direction. Similarly, the magnitudes of the vectors and add to give the magnitude of the resultant vector in the vertical direction. Components along the same axis, say the x-axis, are vectors along the same line and, thus, can be added to one another like ordinary numbers. The same is true for components along the y-axis. (For example, a 9-block eastward walk could be taken in two legs, the first 3 blocks east and the second 6 blocks east, for a total of 9, because they are along the same direction.) So resolving vectors into components along common axes makes it easier to add them. Now that the components of are known, its magnitude and direction can be found.

PDF page 136: , , and shown in the image. Step 2. Find the components of the resultant along each axis by adding the components of the individual vectors along that axis. That is, as shown in Figure 3.30, 3.12 and 3.13

PDF page 137: Step 4. To get the direction of the resultant relative to the x-axis: 3.15 The following example illustrates this technique for adding vectors using perpendicular components.

PDF page 138: and its direction is north of east. Discussion This example illustrates the addition of vectors using perpendicular components. Vector subtraction using perpendicular components is very similar—it is just the addition of a negative vector. Subtraction of vectors is accomplished by the addition of a negative vector. That is, . Thus, the method for the subtraction of vectors using perpendicular components is identical to that for addition. The components of are the negatives of the components of . The x- and y-components of the resultant are thus 3.26 and 3.27 and the rest of the method outlined above is identical to that for addition. (See Figure 3.33.)

PDF page 133: Analytical methods of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. However, analytical methods are more concise, accurate, and precise than graphical methods, which are limited by the accuracy with which a drawing can be made. Analytical methods are limited only by the accuracy and precision with which physical quantities are known.