The Chemistry Maths Book, Second Edition

448 Chapter 16Vectors

16.3 Components of vectors

The component of a vector in a given direction is the

length of its projection in that direction. In Figure

16.8 the component of aalong the direction OP is the

length

ON 1 = 1 |a| 1 cos 1 θ (16.5)

The concept of component is essential for the practical use of vectors for the solution

of physical problems in three dimensions.

We consider first the simpler case of vectors in a plane. Let the initial and terminal

points of ain the xy-plane be(x

1

, y

1

)and(x

2

, y

2

), as shown in Figure 16.9. The

(cartesian) component of the vector in the x-direction is a

x

1 = 1 x

2

1 − 1 x

1

, and the

component in the y-direction isa

y

1 = 1 y

2

1 − 1 y

1

. These two components are sufficient to

specify the vector uniquely. Thus the length of the vector is , and the

direction is given by the slopea

y

2 a

x

. We write the vector in terms of its cartesian

components as

a 1 = 1 (a

x

, a

y

) (16.6)

Also, if the xand ydirections are described by the unit vectors iand j, as in Figure

16.10, then can be expressed as the sum of the two vectors, in the

x-direction (a

x

times the unit vector iin the x-direction) and in the y-direction,

a 1 = 1 a

x

i 1 + 1 a

y

j (16.7)

More generally, a vector in three dimensions (Figure 16.11) can be specified by its

components, a

x

, a

y

and a

z

, in the three cartesian directions i, j, and k(kis the unit

vector in the z-direction).

We write

a 1 = 1 (a

x

, a

y

, a

z

) 1 = 1 a

x

i 1 + 1 a

y

j 1 + 1 a

z

k (16.8)

QP

=a

y

j

OQ

=a

x

i

a=OP

||a=+aa

xy

22

θ

a

o np

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Figure 16.8

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a

a

x

=x

2

−x

1

a

y

=y

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−y

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y

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1

x

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1

y

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p

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Figure 16.9 Figure 16.10