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y 6 4 c b 2 x 6 4 2 2 a 4 6 2 4 d 6
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Figure 4-1
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Four vectors in the Cartesian plane In each case, the magnitude corresponds to the length of the line segment, and the direction is indicated by the arrow
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coordinates, because it allows us to uniquely define any vector as an ordered pair corresponding to The x coordinate of its terminating point (x component) The y coordinate of its terminating point (y component) Figure 4-2 shows the same four vectors as Fig 4-1 does, but all of the originating points have been moved to the coordinate origin The magnitudes and directions of the corresponding vectors in Figs 4-1 and 4-2 are identical That s how we can tell that the vectors a, b, c, and d in Fig 4-2 represent the same mathematical objects as the vectors a, b, c, and d in Fig 4-1
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Cartesian magnitude Imagine an arbitrary vector a in the Cartesian xy plane, extending from the origin (0,0) to the point (xa,ya) as shown in Fig 4-3 The magnitude of a (which can be denoted as ra, as |a|, or as a) can be found by applying the formula for the distance of a point from the origin We learned that formula in Chap 1 Here it is, modified for the vector situation:
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ra = (xa2 + ya2)1/2
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The Cartesian Way
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y 6 4 b a x 6 4 c ( 5, 3) 2 d 4 6 Vectors are denoted as ordered pairs (1, 6) 2 4 6
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(4, 3)
Figure 4-2 These are the same four vectors as shown in
Fig 4-1, positioned so that their originating points correspond to the coordinate origin (0,0)
Magnitude = ra = (xa 2 + ya2 )1/2
(xa, ya) a x xa
Figure 4-3 The magnitude of a vector can be
defined as its length in the Cartesian plane
58 Vector Basics
Figure 4-4 The direction of a vector can be
defined as its angle, in radians, going counterclockwise from the positive x axis in the Cartesian plane
Cartesian direction Now let s think about the direction of a, as shown in Fig 4-4 We can denote it as an angle qa or by writing dir a To define qa in terms of its terminating-point coordinates (xa,ya), we must go back to the polar-coordinate direction-finding system in Chap 3 The following table has those formulas, modified for our vector situation
qa = 0 qa = 0 qa = Arctan ( ya /xa) qa = p /2 qa = p + Arctan ( ya /xa) qa = p qa = p + Arctan ( ya /xa) qa = 3p /2 qa = 2p + Arctan ( ya /xa) When xa = 0 and ya = 0 so a terminates at the origin When xa > 0 and ya = 0 so a terminates on the +x axis When xa > 0 and ya > 0 so a terminates in the rst quadrant When xa = 0 and ya > 0 so a terminates on the +y axis When xa < 0 and ya > 0 so a terminates in the second quadrant When xa < 0 and ya = 0 so a terminates on the x axis When xa < 0 and ya < 0 so a terminates in the third quadrant When xa = 0 and ya < 0 so a terminates on the y axis When xa > 0 and ya < 0 so a terminates in the fourth quadrant
The Cartesian Way
Cartesian vector sum Let s consider two arbitrary vectors a and b in the Cartesian plane, in standard form with terminating-point coordinates
a = (xa,ya) and b = (xb,yb) We calculate the sum vector a + b by adding the x and y terminating-point coordinates separately and then combining the sums to get a new ordered pair When we do that, we get a + b = [(xa + xb),(ya + yb)] This sum can be illustrated geometrically by constructing a parallelogram with the two vectors a and b as adjacent sides, as shown in Fig 4-5 The sum vector, a + b, corresponds to the directional diagonal of the parallelogram going away from the coordinate origin
[(x a+ xb ) , ( ya +y b)]
(x b, y b)