| Excerpt |
|---|
Also known as the vector product, the cross product is a way of multiplying two vectors to yield another vector. |
Use in Physics
In mechanics, the cross product is used in calculating torque and angular momentum. The cross product is also used in introductory electricity and magnetism, where calculations involving the production and effects of magnetic fields generally require the cross product.
Calculating Cross Products
Unit Vector Cross Products
By definition:
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| Wiki Markup |
{excerpt}Also known as the vector product, the cross product is a way of multiplying two vectors to yield another vector.{excerpt} h3. Use in Physics In mechanics, the cross product is used in calculating [torque|torque (single-axis)] and [angular momentum|angular momentum about a single axis]. The cross product is also used in introductory electricity and magnetism, where calculations involving the production and effects of magnetic fields generally require the cross product. h3. Calculating Cross Products h4. Unit Vector Cross Products By definition: {latex}\begin{large}\[\hat{x}\times \hat{y}= \hat{z}\]\end{large}{latex} |
and
...
the
...
same
...
holds
...
for
...
even
...
permutations
...
of
...
the
...
order
...
of
...
the
...
unit
...
vectors,
...
thus:
| Latex |
|---|
}\begin{large}\[ \hat{y} \times \hat{z} = \hat{x} \] \[ \hat{z}\times \hat{x} = \hat{y}\]\end{large}{latex} |
Odd
...
permutations
...
reverse
...
the
...
sign:
| Latex |
|---|
}\begin{large}\[ \hat{y}\times\hat{x} = -\hat{z}\] \[\hat{z}\times\hat{y} = -\hat{x}\] \[\hat{x}\times\hat{z} = -\hat{y}\]\end{large}{latex} {info}For three |
| Info | ||||||||
|---|---|---|---|---|---|---|---|---|
For three dimensions, the sign of the cross product of two unit vectors can be easily remembered by checking if the unit vectors are in a special version of alphabetical order. Start with the position of the {
vector and read to the right. When you get to the end of the equation, wrap to the beginning and keep reading until you return to {
. If you get x, y, z then the sign of the result is positive. If you get x, z, y then the sign is negative. {info} |
and
...
the
...
cross
...
product
...
of
...
any
...
vector
...
with
...
itself
...
is
...
zero:
| Latex |
|---|
}\begin{large}\[ \hat{x}\times\hat{x} = 0\] \[\hat{y}\times\hat{y} = 0\] \[\hat{z}\times\hat{z} = 0\]\end{large}{latex} |
| Note |
|---|
Note that reversing the order of the two vectors being multiplied switches the sign of the result. |
Using this definition, it is possible to find the componentwise cross product of two vectors:
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{note}Note that reversing the order of the two vectors being multiplied switches the sign of the result.{note} Using this definition, it is possible to find the componentwise cross product of two vectors: {latex}\begin{large}\begin{eqnarray*}\vec{A}\times\vec{B}&=&(A_{x}\hat{x}+A_{y}\hat{y}+A_{z}\hat{z})\times(B_{x}\hat{x}+B_{y}\hat{y}+B_{z}\hat{z}) \\ &=& A_{x}B_{x}\hat{x}\times\hat{x} + A_{x}B_{y}\hat{x}\times\hat{y} + A_{x}B_{z}\hat{x}\times\hat{z} + A_{y}B_{x}\hat{y}\times\hat{x} + A_{y}B_{y}\hat{y}\times\hat{y}+A_{y}B_{z}\hat{y}\times\hat{z} \\ & & \qquad\qquad+A_{z}B_{x}\hat{z}\times\hat{x}+A_{z}B_{y}\hat{z}\times\hat{y} + A_{z}B_{z}\hat{z}\times\hat{z}\end{eqnarray*}\end{large}{latex} |
Using
...
the
...
relationships
...
given
...
above
...
for
...
the
...
cross
...
product
...
of
...
unit
...
vectors,
...
we
...
have:
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}\begin{large}\[ A_{x}B_{y}\hat{z} - A_{x}B_{z}\hat{y}-A_{y}B_{x}\hat{z}+A_{y}B_{z}\hat{x} + A_{z}B_{x}\hat{y}-A_{z}B_{y}\hat{x} = (A_{y}B_{z}-A_{z}B_{y})\hat{x} + (A_{z}B_{x} - A_{x}B_{z})\hat{y} +(A_{x}B_{y}-A_{y}B_{x})\hat{z}\]\end{large} |
Shortcut Using Matrix Determinant
One way to remember the formula derived in the section above is to use a matrix determinant:
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{latex} h4. Shortcut Using Matrix Determinant One way to remember the formula derived in the section above is to use a matrix determinant: {latex}\begin{large}\[ \vec{A}\times\vec{B} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ A_{x} & A_{y} & A_{z} \\ B_{x} & B_{y} & B_{z} \end{vmatrix} = (A_{y}B_{z}-A_{z}B_{y})\hat{x} + (A_{z}B_{x} - A_{x}B_{z})\hat{y} +(A_{x}B_{y}-A_{y}B_{x})\hat{z}\]\end{large}{latex} h3. Geometric Methods h4. Magnitudes from Trigonometry The formalism above has a simple geometric interpretation. The cross product measures the "perpendicularity" of two vectors. Since Cartesian unit vectors are always either perpendicular ( |
Geometric Methods
Magnitudes from Trigonometry
The formalism above has a simple geometric interpretation. The cross product measures the "perpendicularity" of two vectors. Since Cartesian unit vectors are always either perpendicular (
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{latex}\begin{large}$\hat{x}\perp \hat{y}, \hat{z}$\end{large}{latex} |
)
...
or
...
parallel
...
(
| Latex |
|---|
}\begin{large}$\hat{x} \parallel \hat{x}$\end{large}{latex} |
)
...
we
...
get
...
a
...
cross
...
product
...
with
...
either
...
magnitude
...
one
...
(for
...
perpendicular
...
unit
...
vectors)
...
or
...
zero
...
(for
...
parallel
...
unit
...
vectors).
...
The
...
mathematical
...
definitions
...
given
...
above,
...
however,
...
will
...
let
...
you
...
construct
...
cross
...
products
...
with
...
vectors
...
that
...
are
...
combinations
...
of
...
the
...
unit
...
vectors,
...
such
...
as
| Latex |
|---|
}\begin{large}$\vec{A} = \frac{1}{\sqrt{2}}\hat{x} + \frac{1}{\sqrt{2}}\hat{y}$\end{large}{latex} |
.
...
Two
...
arbitrary
...
vectors
...
will
...
usually
...
not
...
be
...
perfectly
...
parallel
...
or
...
perpendicular.
...
Instead,
...
they
...
will
...
form
...
some
...
angle
...
θ as
...
shown
...
in
...
the
...
figures
...
below.
By using the mathematical definition, it is possible to show that for the case of two vectors A and B that meet at an angle θ, the magnitude of the cross product will be:
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!VecAngle.png|height=150! By using the mathematical definition, it is possible to show that for the case of two vectors *A* and *B* that meet at an angle θ, the magnitude of the cross product will be: {latex}\begin{large}\[ |\vec{A}\times \vec{B}| = |A||B|\sin\theta \]\end{large}{latex} {tip}Note that this definition gives a magnitude of one for the product of two perpendicular unit vectors and a magnitude of zero for two parallel unit vectors.{tip} h4. Magnitudes from Vector Parallelograms When manipulating vectors, it is sometimes useful to imagine the parallelogram constructed by adding the two vectors in both possible orders |
| Tip |
|---|
Note that this definition gives a magnitude of one for the product of two perpendicular unit vectors and a magnitude of zero for two parallel unit vectors. |
Magnitudes from Vector Parallelograms
When manipulating vectors, it is sometimes useful to imagine the parallelogram constructed by adding the two vectors in both possible orders (e.g.,
...
A
...
+
...
B
...
and
...
B
...
+
...
A
...
).
...
The
...
magnitude
...
of
...
the
...
sum
...
of
...
the
...
two
...
vectors
...
can
...
then
...
be
...
interpreted
...
as
...
the
...
length
...
of
...
the
...
diagonal
...
of
...
the
...
parallelogram.
...
The
...
cross
...
product
...
can
...
be
...
similarly
...
interpreted.
...
The
...
magnitude
...
of
...
the
...
cross
...
product
...
of
...
two
...
vectors
...
is
...
equal
...
to
...
the
...
area
...
of
...
the
...
vector
...
parallelogram.
Direction from Right Hand Rule
We have given two geometric interpretations of the size of the cross product. Unfortunately, the direction of the cross product is not similarly meaningful. Consider a comparison between vector addition and the cross product. Vector addition is commutative, which means that if A + B = C, then it is also true that B + A = C. For this reason, the preferred direction of the diagonal of the vector parallelogram is unambiguous. The diagonal should point "along with" the arrows of the sides.
The preferred direction for the cross product is not obvious in the same way. One signal of the difficulty is that the cross product is not commutative. If
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!VecParallelogram.png|height=150! h4. Direction from Right Hand Rule We have given two geometric interpretations of the size of the cross product. Unfortunately, the direction of the cross product is not similarly meaningful. Consider a comparison between vector addition and the cross product. Vector addition is commutative, which means that if *A* + *B* = *C*, then it is also true that *B* + *A* = *C*. For this reason, the preferred direction of the diagonal of the vector parallelogram is unambiguous. The diagonal should point "along with" the arrows of the sides. !VecAddCross.png|height=150! The preferred direction for the cross product is not obvious in the same way. One signal of the difficulty is that the cross product is not commutative. If {latex}\begin{large}$\vec{A}\times\vec{B} =\vec{C}$\end{large}{latex} |
,
...
then
...
our
...
mathematical
...
definition
...
tells
...
us
...
that
| Latex |
|---|
}\begin{large}$\vec{B}\times\vec{A} = - \vec{C}$\end{large}{latex} |
.
...
The
...
order
...
of
...
the
...
vectors
...
in
...
the
...
equation
...
matters
...
for
...
determining
...
the
...
direction
...
of
...
the
...
cross
...
product.
...
This
...
difficulty
...
shows
...
up
...
in
...
the
...
geometric
...
interpretation
...
of
...
the
...
cross
...
product
...
by
...
noticing
...
that
...
if
...
we
...
define
...
the
...
direction
...
of
...
the
...
cross
...
product
...
to
...
be
...
perpendicular
...
to
...
the
...
surface
...
of
...
the
...
parallelogram,
...
there
...
are
...
two
...
equally
...
good
...
choices
...
.
...
If
...
we
...
construct
...
a
...
parallelogram
...
that
...
lies
...
in
...
the
...
x,y
...
plane,
...
for
...
example,
...
then
...
either
...
the
...
+z
...
or
...
-z
...
direction
...
is
...
perpendicular
...
to
...
the
...
parallelogram.
| Info |
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| } The situation is even worse if we tried to define a direction for the area that wasn't perpendicular to the parallelogram. Instead of only two directions, we would have an infinite number to choose from! {info} |
The
...
fact
...
is
...
that
...
the
...
only
...
way
...
to
...
define
...
a
...
direction
...
for
...
the
...
cross
...
product
...
is
...
to
...
make
...
an
...
arbitrary
...
rule.
...
The
...
rule
...
has
...
already
...
been
...
incorporated
...
in
...
the
...
mathematical
...
definition
...
we
...
gave
...
above.
...
The
...
definition
...
we
...
stated
...
makes
...
the
...
product
| Latex |
|---|
}\begin{large}$\hat{x}\times\hat{y}$\end{large}{latex} |
equal
...
to
...
plus
| Latex |
|---|
}\begin{large}$\hat{z}$\end{large}{latex} |
rather
...
than
...
minus
| Latex |
|---|
}\begin{large}$\hat{z}$\end{large}{latex} |
.
...
This
...
was
...
an
...
arbitrary
...
choice,
...
based
...
on
...
the
...
traditional
...
ordering
...
of
...
those
...
unit
...
vectors.
...
Once
...
that
...
choice
...
has
...
been
...
made,
...
all
...
we
...
need
...
is
...
a
...
simple
...
rule
...
to
...
remember
...
the
...
consequences.
...
The
...
most
...
widely
...
taught
...
mnemonic
...
is
...
the
...
"right
...
hand
...
rule".
...
To
...
find
...
the
...
direction
...
of
...
the
...
cross
...
product
...
of
...
two
...
vectors,
...
start
...
by
...
carefully
...
reading
...
the
...
order
...
of
...
the
...
vectors.
...
For
| Latex |
|---|
}\begin{large}$\vec{A}\times\vec{B}$\end{large}{latex} |
,
...
begin
...
by
...
laying
...
the
...
fingers
...
of
...
your
...
right
...
hand
...
along
...
vector
...
A
...
(the
...
first
...
in
...
the
...
product).
...
Then,
...
curl
...
your
...
fingers
...
toward
...
B
...
.
...
Your
...
thumb
...
will
...
indicate
...
the
...
direction
...
of
...
the
...
product
...
vector.
| Warning |
|---|
You will get the wrong answer if you use your left hand. |
| Tip |
|---|
Check that this definition reverses the direction of the product if you start with your fingers along B and curl toward A. !VecCrossProduct.png|height=200! {warning}You will get the *wrong answer* if you use your left hand.{warning} {tip}Check that this definition reverses the direction of the product if you start with your fingers along *B* and curl toward *A*.{tip} |