Vectors - Rotation Transforms for Computer Graphics - page 20

3.6.1 Multiplying a Vector by a Scalar

Given a vector n ,2 n means that the vectors components are doubled. For example,
given

T T .

n =[ 345 ] then 2 n =[ 6810 ]

Similarly, dividing n by 2, its components are halved. Note that the vector's direction
remains unchanged - only its magnitude changes. However, the vector's direction is
reversed if the scalar is negative:

T .

λ =− 2 then λ n =[− 6 − 8 − 10 ]

In general, given

⎡ ⎤ ⎡ ⎤

n 1
n 2
n 3

± λn 1
± λn 2
± λn 3

⎣ ⎦ ⎣ ⎦

n = then ± λ n = where λ is a scalar.

3.6.2 Vector Addition and Subtraction

Given vectors r and s , r ± s is defined as

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

x r
y r
z r

x s
y s
z s

x r ± x s

⎣ ⎦ , ⎣ ⎦ ⎣ ⎦ .

r = s = then r ± s = y r ± y s

z r ± z s

Vector addition is commutative:

a + b = b + a

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

1
2
3

4
5
6

4
5
6

1
2
3

⎣ ⎦ + ⎣ ⎦ = ⎣ ⎦ + ⎣ ⎦ .

e.g.

However, like scalar subtraction, vector subtraction is not commutative

a − b = b − a

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

4
5
6

1
2
3

1
2
3

4
5
6

⎣ ⎦ − ⎣ ⎦ = ⎣ ⎦ − ⎣ ⎦ .

e.g.

3.7 Position Vectors

Given any point P(x,y,z) ,a position vector p is created by assuming that P is
the vector's head and the origin is its tail. Because the tail coordinates are ( 0 , 0 , 0 )
the vector's components are x,y,z . Consequently, the vector's length | p | equals