Vector Components

Figure 10 Vector Addition Not in Same Line

Classical Physics - Physics theories and other related topics
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VECTORS: RESULTANTS AND COMPONENTS Vectors CP-02 Page 10 Rev. 0 It is left as an exercise for the student to show that vector addition is commutative, using the above example. Specifically, make a scale drawing showing that traveling 3 miles north and then 4 miles east yields the same resultant as above. It is also reasonably obvious that more than two vectors can be added. One can travel three miles east and then three miles north and then three miles west and arrive at a point three miles north of the starting point. The sum of these three displacements is a resultant displacement of three miles north. (If this is not immediately apparent, sketch it.) A student problem is to find the net or resultant displacement if a person travels 9 miles south and then 12 miles east and then 25 miles north. Make a scale drawing and determine the magnitude and direction of the resultant displacement. A scale of 2 miles per centimeter or 4 miles per inch will fit the drawing on standard paper. Answer: About 20 miles at 53 north of east. o Vector Components Components of a vector are vectors, which when added, yield the vector. For example, as shown in the previous section (Figure 10), traveling 3 miles north and then 4 miles east yields a resultant displacement of 5 miles, 37 north of east. This example demonstrates that o component vectors of any two non-parallel directions can be obtained for any resultant vector in the same plane. For the purposes of this manual, we restrict our discussions to two dimensional space. The student should realize that vectors can and do exist in three dimensional space. One could write an alternate problem: "If I am 5 miles from where I started northeast along a line 37 N of east, how far north and how far east am I from my original position?" Drawing this o on a scale drawing, the vector components in the east and north directions can be measured to be about 4 miles east and 3 miles north. These two vectors are the components of the resultant vector of 5 miles, 37 north of east. o Component vectors can be determined by plotting them on a rectangular coordinate system. For example, a resultant vector of 5 units at 53 can be broken down into its respective x and y o magnitudes. The x value of 3 and the y value of 4 can be determined using trigonometry or graphically. Their magnitudes and position can be expressed by one of several conventions including: (3,4), (x=3, y=4), (3 at 0 , 4 at 90 ), and (5 at 53 ). In the first expression, the first o o o term is the x-component (F ), and the second term is the y-component (F ) of the associated x y resultant vector.

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