Understanding Engineering Mathematics

x

O

i

j

k

y

z

Figure 11.10Basic vectorsi,j,k.

and this provides the link with the ‘arrows’ representation of vectors given in earlier
sections. It is common practice to representa 1 i+a 2 j+a 2 kby a triple(a 1 ,a 2 ,a 2 )or a
‘column matrix’

a=

[a

1

a 2

a 3

]

but always remember that these only represent the vectorawith respect to a particular
set of basis vectors, and should not be confused with the notation for the coordinates of
a point.
Any set of unit vectors, such asi,j,kwhich are mutually perpendicular to each other
(and there can be at most three in 3-dimensional space) is called anorthonormal set.The
i,j,kform anorthonormal basisfor the set of all vectors, in the sense that any three
dimensional vector can be expressed in terms of them.
Note thata 1 i+a 2 j+a 3 k= 0 canonlymeana 1 =a 2 =a 3 =0.

Problem 11.3

Finda,b,cif.aYbYc/iY.b−cY 1 /jY.aYc/k=0.

Don’t let the Greek symbols put you off – as mathematics gets more advanced we soon
run out of alphabets and so you will have to get used to such notation.
If(α+β+χ)i+(β−χ+ 1 )j+(α+χ)k= 0 then

α+β+χ= 0 (i)

β−χ+ 1 = 0 (ii)

α+χ= 0 (iii)

So a single vector equation is equivalent to three ‘scalar’ equations. We will consider such
systems of equations in detail in Chapter 13, but the above system is not difficult to solve.
Substituting forβfrom (ii) into (i) gives

α+ 2 χ= 1

α+χ= 0

from whichχ=1,α=−1, thenβ=χ− 1 =0.