Game Development Reference

In-Depth Information

The vector
V
in Equation (2.1) may also be represented by a matrix having a

single column and
n
rows:

V

V

1

2

V

=

.

(2.2)

V

n

We treat this column vector as having a meaning identical to that of the comma-

separated list of components written in Equation (2.1). Vectors are normally ex-

pressed in these forms, but we sometimes need to express vectors as a matrix

consisting of a single row and
n
columns. We write row vectors as the transpose

of their corresponding column vectors:

]

T

[

V

=

VV

.

V

(2.3)

1

2

n

A vector may be multiplied by a scalar to produce a new vector whose com-

ponents retain the same relative proportions. The product of a scalar
a
and a vec-

tor
V
is defined as

a

VV

==

a

aV

,

aV

,

.

,

aV

(2.4)

1

2

n

In the case that

a

=−

1

, we use the slightly simplified notation

V
to represent the

−

negation of the vector
V
.

Vectors add and subtract componentwise. Thus, given two vectors
P
and
Q
,

we define the sum

PQ
as

+

PQ

+= +

PQP Q

,

+

,

,

P Q

+

.

(2.5)

1

1

2

2

n

n

The difference between two vectors, written

Q
, is really just a notational sim-

P

−

(

)

plification of the sum

PQ
.

With the above definitions in hand, we are now ready to examine some fun-

damental properties of vector arithmetic.

+−

Theorem 2.1.
Given any two scalars
a
and
b
, and any three vectors
P
,
Q
, and

R
, the following properties hold.

(a)

P

+=+

QQP

(

)

(

)

(b)

PQ R P QR

++=++

Search Nedrilad ::

Custom Search