Cambridge Additional Mathematics

Vectors (Chapter 11) 287

SCALAR MULTIPLICATION

Ascalaris a non-vector quantity. It has a size but no direction.

We can multiply vectors by scalars such as 2 and¡ 3 , or in fact any k 2 R.

Ifais a vector, we define 2 a=a+a and 3 a=a+a+a

so ¡ 3 a=3(¡a)=(¡a)+(¡a)+(¡a).

Ifais then

So, 2 a is in the same direction asabut is twice as long asa

3 a is in the same direction asabut is three times longer thana

¡ 3 a has the opposite direction toaand is three times longer thana.

Ifais a vector andkis a scalar, thenkais also a vector and we are

performingscalar multiplication.

If k> 0 , kaandahave the same direction.

If k< 0 , kaandahave opposite directions.

If k=0, ka= 0 , the zero vector.

Ifkis any scalar and v=

μ

v 1

v 2

¶

, then kv=

μ

kv 1

kv 2

¶

.

Notice that:

² (¡1)v=

μ

(¡1)v 1

(¡1)v 2

¶

=

μ

¡v 1

¡v 2

¶

=¡v ² (0)v=

μ

(0)v 1

(0)v 2

¶

=

μ

0

0

¶

= 0

Example 6 Self Tutor

If p=

μ

4

1

¶

and q=

μ

2

¡ 3

¶

, find: a 3 q b p+2q c^12 p¡ 3 q

a 3 q

=3

μ

2

¡ 3

¶

=

μ

6

¡ 9

¶

b p+2q

=

μ

4

1

¶

+2

μ

2

¡ 3

¶

=

μ

4 + 2(2)

1+2(¡3)

¶

=

μ

8

¡ 5

¶

c^12 p¡ 3 q

=^12

μ

4

1

¶

¡ 3

μ

2

¡ 3

¶

=

μ 1

2 (4)¡3(2)

1

2 (1)¡3(¡3)

¶

=

μ

¡ 4

(^912)
¶
VECTOR SCALAR
MULTIPLICATION
a
a
2 a a
a
a
3 a

• a
  
  
  • a
    
    
    • a
    
    
    
    
  
  
  
  

-3a

a

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_11\287CamAdd_11.cdr Monday, 6 January 2014 9:55:37 AM BRIAN