16.2 Vector algebra 445
Figures 16.2 (a) and (b) illustrate several types of vector associated with the motion
of a body in space, along curve PQ, under the influence of external forces. In Figure
(a), the position of point A on the curve is given by whose initial point is at
the origin of a coordinate system and whose terminal point is at A. The vector ais
called a position vector, and its value (magnitude and direction) changes as the body
moves along the curve. When the body has moved from A to B the position vector has
changed from ato b, and the displacement A to B is given by the vector. We
will see that vector algebra givesc 1 = 1 b 1 − 1 a; that is, the displacement A to B is equal to
the change of position vector. We note that a position vector is a type of displacement
vector; but one whose initial point is bound to a fixed point, the origin of a coordinate
system.
At each point along the curve PQ, the body has velocity valong the direction of
motion at that point; that is, the direction of vat every point is tangential to the curve.
Figure (b) shows the velocities at points A and B, and also forces acting on the body at
these points.
16.2 Vector algebra
Equality
Two vectors aand bare equal,
a 1 = 1 b (16.1)
if they have the same length and the same direction. The initial points of equal vectors
need coincide onlyif they are bound vectors, such as position vectors. Thus two separated
bodies moving at the same speed and in the same direction have equal velocities.
Vector addition
The sum, or resultant,a 1 + 1 bof vectors aand bis defined geometrically in Figure 16.3.
In (a) the vectors are positioned with the initial point of bcoincident with the
terminal point of a. The suma 1 + 1 bis then equal to the vector drawn from the initial
point of ato the terminal point of b. Figure (b) also shows that addition is commutative,
with
a 1 + 1 b 1 = 1 b 1 + 1 a (16.2)
c=AB
a=OA
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(b)
Figure 16.2