Calculus: Analytic Geometry and Calculus, with Vectors

50 Vectors and geometry in three dimensions

The sum r of two vectors u and v is,*s-in-Figttre-2-4-3, the vector which

runs from the tail of u to the head of v when the tail of v isplaced at the

Figure 2.13

nl

Figure 2.131

head of u. The figure shows that

v + u = u + v. Because the sum

of two vectors is, as in Figure 2.13,

the diagonal of a parallelogram, the

rule (or law) for addition of vectors

is called the parallelogram law. Fig-

ure 2.131 shows the sum r of four

vectors u1, u2, ua, n4. In applied

mathematics the sum of two or more

vectors is sometimes called their resultant.

The difference u - v is defined to be the sum of u and -v, so that

u-v

Figure 2.14

u - v = u + (-v). The most obvious way to find

u - v is to find -v and add it to U. In substantially

all cases, it is quicker, easier, and more useful to observe

that u - v is the vector which we must add to v to

obtain the sum u. When the tails of u and v coincide,

the vector u - v runs from the head of v to the head of u.

It is worthwhile to look at the italicized statement and

Figure 2.14 until both are thoroughly understood and

remembered. The figure clearly says that

(2.141) u= v+ (U -V).

Since angles between vectors can be sources of confusion and mis-

understanding, we give a little careful attention to the subject. In case

one or the other of two vectors has length 0, there is no reasonable way to

determine an angle that should be called the angle between them, and we

say that the angle is undetermined or undefined. Two sharpened pencils

intersect, we can choose any point 0 in E3 and replace

the vectors by equal vectors having their tails at 0

as in Figure 2.15. Suppose first that these vectors u

of positive length represent vectors in the directions

of their sharpened tips. In case these vectors do not

o- and v have neither the same nor opposite directions.

Figure 2.15 These vectors then determine the plane in which they

lie. The angle 0 is determined by the method used in

trigonometry to introduce radian measure. The first step is to draw, in

the plane of the vectors, a circle of radius a with center atO and to find the

length s of the shorter of the two arcs into which the vectors cut the circle.

The number 0 defined by

(2.151) length of arc

0 =

s

or angle =

a radius