Vectors - 3D Math Primer for Graphics and Game Development - page 38

not “turn” between steps, so “forward” is always parallel to +z.) This
displacement is illustrated in Figure 2.5.
The order in which we perform the steps is not important; we could
move 4 units forward, 3 units down, and then 1 unit to the right, and
we would have displaced by the same total amount. The different order-
ings correspond to different routes along the axially aligned bounding box
containing the vector. Section 2.7.2 mathematically verifies this geometric
intuition.

2.3.2 The Zero Vector

For any given vector dimension, there is a special vector, known as the zero
vector, that has zeroes in every position. For example, the 3D zero vector is
[0,0,0]. We denote a zero vector of any dimension using a boldface zero: 0 .
In other words,

2
4 0

3

5

The zero vector

0
.
0

0 = .

The zero vector is special because it is the only vector with a magnitude
of zero. All other vectors have a positive magnitude. The zero vector is
also unique because it is the only vector that does not have a direction.
Since the zero vector doesn't have a direction or length, we don't draw
it as an arrow like we do for other vectors. Instead, we depict the zero
vector as a dot. But don't let this make you think of the zero vector as a
“point” because a vector does not define a location. Instead, think of the
zero vector as a way to express the concept of “no displacement,” much as
the scalar zero stands for the concept of “no quantity.”
Like the scalar zero you know, the zero vector of a given dimension is
the additive identity for the set of vectors of that dimension. Try to take
yourself back to your algebra class, and retrieve from the depths of your
memory the concept of the additive identity: for any set of elements, the
additive identity of the set is the element x such that for all y in the set,
y+x = y. 2 In other words, when we add the zero vector to any other vector,
we get that vector: 0 + a = a . Section 2.7 deals with vector addition.

2 The typeface used here is not intended to limit the discussion to the set of scalars.
We are talking about elements in any set. Also, we request leniency from the abstract
algebra sticklers for our use of the word “set,” when we should use “group.” But the
latter term is not as widely understood, and we could only afford this footnote to dwell
on the distinction.