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Unit Vector Notation
1 Definitions
Figure 1 shows the definition of components of a vector
V. The angles α, β and γ are the angles
that the vector makes to the three coordinate axes x, y and z respectively. The dashed lines are
perpendiculars drawn from the tip of the vector to the three coordinate axis. The components
V
x
, V
y
and V
z
of
V are defined to be the segments of the coordinate axes marked out by these
perpendiculars as shown. Hence, using the definition of cosine for the three right-angle triangles,
we find,
V
x
= V cos α, V
y
= V cos β, V
z
= V cos γ,
where V (short for |
V|) is the magnitude of the vector
V. We also define unit vectors (vectors
of magnitude one) along each of the three coordinate axes x, y and z to be
i,
j and
k respectively
(figure 2). The following three vectors in the three coordinate directions can now be defined.
V
x
= V
x
i,
V
y
= V
y
j,
V
z
= V
z
k.
Using the triangle rule for vector addition twice, this gives,
V =
V
x
V
y
V
z
= V
x
i + V
y
j + V
z
k.
This is known as the unit vector notation of a vector.
If the vector is restricted to the x-y plane, then γ = 90
. This makes (figure 3),
V
z
= 0, and α + β = 90
Hence,
cos β = cos (
− α) = sin α.
Thus the components V
x
and V
y
can now be written as,
V
x
= V cos α, V
y
= V sin α
x
y
z
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~
V
V
x
V
y
V
z
α
β
γ
Figure 1: Components of a vector
~
V
x
y
z
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ˆ
i
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j
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ˆ
k
Figure 2: Unit vectors
ˆ
i,
ˆ
j and
ˆ
k along the three coordinate directions.
x
y
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~
V
V
x
V
y
α
β
Figure 3: Components of a vector
~
V restricted to the x-y plane (β = 90
− α).
θ
(
~
A ×
~
B)
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B
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A
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~
B ..
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~
A
(
~
A ×
~
B)
thumb out of page
curled fingers
Figure 4: Finding the direction of the cross product
~
A ×
~
B using the right-hand-rule. Note that
the symbol means out of the page and ⊗ means into the page.
definition uses the so-called right-hand-rule. If you put the four fingers of the right hand
together and curl them along the angle θ going from the first vector of the product (
~
A in this
case) towards the second vector of the product (
~
B in this case), the direction in which the
thumb will stick out is the direction of the cross product (see figure 4).
To compute cross products of vectors given in a unit vector notation, it is useful to know the cross
products of the individual unit vectors
ˆ
i,
ˆ
j and
ˆ
k. From the above definition, it is straightforward
to see the following.
ˆ
i ×
ˆ
j = −
ˆ
j ×
ˆ
i =
ˆ
k,
ˆ
j ×
ˆ
k = −
ˆ
k ×
ˆ
j =
ˆ
i,
ˆ
k ×
ˆ
i = −
ˆ
i ×
ˆ
k =
ˆ
j. (2)
Also
ˆ
i ×
ˆ
i =
ˆ
j ×
ˆ
j =
ˆ
k ×
ˆ
k = 0. (3)
Note that the definition of the cross product requires a minimum of three dimensions. So,
drawing figures on a two-dimensional page becomes tricky. Hence, we shall often use a convention
for representing directions out of the page and into the page. The symbol means out of the page
and the symbol ⊗ means into the page. To remember these symbols, it is convenient to think of
them as the tip (point) and the tail (feathers) of an actual arrow.
