The Chemistry Maths Book, Second Edition

446 Chapter 16Vectors

In this case aand bdefine the pairs of parallel sides of a parallelogram, and vector

addition is said to obey the parallelogram law. If aand bare displacements of a body

then their suma 1 + 1 bis the total displacement. If aand bare two forces acting on a

body, thena 1 + 1 bis the total (or resultant) force; the parallelogram law is then called

the ‘parallelogram of forces’.

2

Subtraction

Ifa 1 + 1 b 1 = 10 , where 0 is the null vector, thenb 1 = 1 −ais a vector

that has the same length as a,|b| 1 = 1 |a|, but points in the

opposite direction. The subtraction of vectors is then defined

by (Figure 16.4)

a 1 + 1 (−b) 1 = 1 a 1 − 1 b (16.3)

Scalar multiplication

The vectora 1 + 1 a 1 = 12 ahas twice the length of aand has the same direction. In general,

the product of a scalar (number) cand a vector ais written as ca. It has ctimes the

length of a, has the same direction as aif cis positive, and has opposite direction if c

is negative (Figure 16.5).

Ifc 1 = 10 thenca 1 = 10 , and the direction is not defined. A vector adivided by its length

|a|is the unit vector âthat has the same direction as a:

(16.4)

Unit vectors are often used to define direction.

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(b)

Figure 16.3

2

The parallelogram of velocities was described by Heron of Alexandria (1st century AD) in his Mechanics, but

also appears in a work attributed to Aristotle (384–322 BC).

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Figure 16.4

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Figure 16.5