Cambridge Additional Mathematics

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292 Vectors (Chapter 11)

are parallel vectors of different length.

Two non-zero vectors areparallelif and only if one is a scalar multiple of the other.

Given any non-zero vectoraand non-zero scalark, the vectorkais parallel toa.

² Ifais parallel tob, then there exists
a scalarksuch that a=kb.
² If a=kb for some scalark, then
I ais parallel tob, and
I jaj=jkjjbj.

Example 10 Self Tutor


Findrgiven that a=

μ
¡ 1
r


is parallel to b=

μ
2
¡ 3


Sinceaandbare parallel, a=kb for some scalark.

)

μ
¡ 1
r


=k

μ
2
¡ 3


) ¡1=2k andr=¡ 3 k
) k=¡^12 and hence r=¡3(¡^12 )=^32

UNIT VECTORS


Given a non-zero vectora, its magnitudejajis a scalar quantity.

If we multiplyaby the scalar
1
jaj
, we obtain the parallel vector
1
jaj
a with length 1.

² A unit vector in the direction ofais
1
jaj

a.

² A vector of lengthkin the same direction asais

k
jaj

a.

² A vector of lengthkwhich isparallel toacould be §

k
jaj

a.

E Parallelism


jj
jj

kkis the modulus of ,
whereas is the length
of vector.

a
a

a^2 a Qea

a
b

cyan magenta yellow black

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Additional Mathematics
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