A vector can be multiplied by a scalar. The components of the
vector are multiplied by the scalar and the result is a scaled vector which in
the same direction as the original vector if the scalar is positive, or in the
opposite direction if the scalar is negative.

A vector can also be multiplied by another vector. Two types of vector
multiplications have been defined, the ** scalar product** and the
** vector product**.

The **scalar product** or **dot product**
**A×B **of two vectors **A** and **B**
is not a vector, but a scalar quantity (a number with units). In terms of the Cartesian components of the vectors **A** and **B** the
scalar product is written as

**A×B **= A_{x}B_{x }+
A_{y}B_{y }+ A_{z}B_{z}.

Consider two arbitrary vectors **A** and **B** and choose
the orientation of your Cartesian coordinate system such that **A** points
into the x-direction and **B** lies in the x-y plane. Then **A** =
(A_{x}, 0, 0) and **B **= (B_{x}, B_{y}, 0) and

**A×B **= A_{x}B_{x}.

Since A_{x}=A and B_{x}= Bcosf
we can also write

**A×B **= ABcosf
.

The scalar product of two vectors **A** and **B**
is a scalar quantity equal to the product of the
magnitudes of the two vectors and the cosine of the smallest angle between them.
The scalar product is commutative,

**A×B **= **B×A**.

When we form the scalar product of two vectors, we multiply the **parallel
component** of the two vectors.

Many physical quantities of interest are calculated by forming the scalar
product of two vectors.

Examples:

To find the components of any vector quantity **A** along some arbitrary
direction defined by the spherical
angles q and f, form
the unit vector

**u** = (u_{x}, u_{y}, u_{z}) =(sinq cosf,
r
sinq
sinf, r cosq)

and
then evaluate **A×u **= A_{x}u_{x
}+
A_{y}u_{y }+ A_{z}u_{z}.

The **vector product** or "**cross
product**" of two vectors **A** and **B**
is a vector **C**, defined as **C**=**A**´**B.**

We can find the Cartesian components of **C**=**A**´**B**
in terms of the components of **A** and **B**.

C_{x}=A_{y}B_{z}-A_{z}B_{y}

C_{y}=A_{z}B_{x}-A_{x}B_{z}

C_{z}=A_{x}B_{y}-A_{y}B_{x }

Again, consider two arbitrary vectors **A** and **B** and choose the
orientation of your Cartesian coordinate system such that **A** points into
the x-direction and **B** lies in the x-y plane. Then **A** = (A_{x},
0, 0) and **B **= (B_{x}, B_{y}, 0) and

C_{x}=0

C_{y}=0

C_{z}=A_{x}B_{y}.

The magnitude of **C** is

C=C_{z}=A_{x}B_{y}.

Since A_{x}=A and B_{y}= Bsinf we
can also write

C=ABsinf,

where f is the smallest angle between the directions of the vectors **A** and **B**.
**C** is perpendicular to both **A** and **B**, i.e. it is
perpendicular to the plane that contains both **A** and **B**. The
direction of **C** can be found by inspecting its components or by using the**
right-hand rule**.

Let the fingers of your right hand point in the direction of **A**.
Orient the palm of your hand so that, as you curl your fingers, you can sweep
them over to point in the direction of **B**. Your thumb points in the
direction of **C**=**A**´**B**.

If **A** and **B** are parallel or anti-parallel to each other, then **C**=**A**´**B**=0,
since sinf=0. If **A** and **B** are
perpendicular to each other, then sinf=1 and **C**
has its maximum possible magnitude.

When we form the scalar product of two vectors, we multiply the **perpendicular
component** of the two vectors. The vector product is not commutative,

**A**´**B **= -**B**´**A**.

Many physical quantities of interest are calculated by forming the vector product of two vectors.

Examples:

A **torque ****t**
is the product of a lever arm and a force that is applied perpendicular to the
lever arm. It is the vector product of **r** and **F**, **t**=**r**´**F**.

The **angular momentum** **L**
of the particle about a point is **L**=**r**´**p**,
where **r** is the displacement vector of the particle from the point and **p**
is its momentum. Only the component of **p** perpendicular to **r**
contributes to the angular momentum **L**.

Look at this **3D
visualization** of the cross product.

**Vector Formulas:**

**A×**(**B**´**C**)=**B×**(**C**´**A**)=**C×**(**A**´**B**)

**A**´(**B**´**C**)=**B**(**A×**)=**C**(**A×**)
(bac-cab rule)

(**A**´**B**)**×**(**B**´**C**)=(**A×**)(**B×**)-(**A×**)(**B×**)

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