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EXAMPLE 2
Representing the sum and difference of two algebraic vectors in
Given and , determine the components
of a ! a !
b ! OA !
and a ! 11,
b ! 32 OB ! b ! 14, 22
, and illustrate each of these vectors on the graph.
R 2
Solution
D(–3, 5)
4
y
A(1, 3)
a – b
2
a
C(5, 1)
–5
0
–2
a – b
b
a + b x
5
B(4, –2)
a ! b ! OA ! OB ! 11, 32 14, 22 11 4, 3 1222 15, 12 OC !
a ! b ! OA ! OB ! 11, 32 14, 22 11 4, 3 22 13, 52 OD !
From the diagram, we can see that a ! b !
and BA !
represent the diagonals of the
parallelogram. It should be noted that the position vector, , is a vector that
is equivalent to diagonal BA ! . The vector OD ! a ! b ! OD !
is described as a position
vector because it has its tail at the origin and is equivalent to BA !
, since their
magnitudes are the same and they have the same direction.
Vectors in R 2 Defined by Two Points
In considering the vector AB !
y
, determined by the points A1x 1 , y 1 2 and
B1x 2 , y 2 2, an important consideration is to be able to find its related
position vector and to calculate @ AB ! @ . In order to do this, we use the
B(x
triangle law of addition. From the diagram on the left,
and AB ! OB ! OA ! OA ! AB ! OB ! 2 , y 2 )
A(x 1 , y 1 )
,
P(x 2 – x 1 , y 2 – y 1 )
1x 2 , y 2 2 1x 1 , y 1 2 1x 2 x 1 , y 2 y 1 2.
x
Thus, the components of the algebraic vector are found by subtracting
0
the coordinates of its tail from the coordinates of its head.
To determine @ AB ! @ , use the Pythagorean theorem.
OP is the position vector
@ AB ! for AB.
@ V1x 2 x 1 2 2 1y 2 y 1 2 2
The formula for determining @ AB ! @ is the same as the formula for
finding the distance between two points.
322 6.6 OPERATIONS WITH ALGEBRAIC VECTORS IN R 2
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