You might be already familiar with the concept of performing a number of various **operations** between vectors, so in this course, let us review some essential operations that are relevant to start working with quantum mechanics:
!!! info "Addition"
I can add two vectors to produce a third vector, $$\vec{a} + \vec{b}= \vec{c}$$.
As with scalar addition, also vectors satisfy the commutative property, $$\vec{a} + \vec{b} = \vec{b} + \vec{a}$$.
Vector addition can be carried out in terms of their components,
In addition to multiplying a vector by a scalar, as mentioned above, one can also multiply two vectors among them.
There are two types of vector productions, one where the end result is a scalar (so just a number) and the other where the end result is another vectors.
In addition to multiplying a vector by a scalar, as mentioned above, one can also multiply two vectors among them.
There are two types of vector productions, one where the end result is a scalar (so just a number) and the other where the end result is another vectors.
!!! info "The scalar production of vectors"
The **scalar production of vectors is given by $$ \vec{a}\cdot \vec{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \, .$$
Note that since the scalar product is just a number, its value will not depend on the specific
basis in which we express the vectors: the scalar product is said to be *basis-independent*. The scalar product is also found via
$$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos \theta$$ with $\theta$ the angle between the vectors.
The **scalar production of vectors is given by $$ \vec{a}\cdot \vec{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \, .$$
Note that since the scalar product is just a number, its value will not depend on the specific
basis in which we express the vectors: the scalar product is said to be *basis-independent*. The scalar product is also found via
$$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos \theta$$ with $\theta$ the angle between the vectors.
!!! info "Cross product"
"The vector product (or cross product) between two vectors $\vec{a}$ and $\vec{b}$ is given by
where $|\vec{a}|=\sqrt{ \vec{a}\cdot\vec{a} }$ (and likewise for $|\vec{b}|$) is the norm of the vector $\vec{a}$, $\theta$ is the angle between the two vectors, and $\hat{n}$ is a unit vector which is *perpendicular* to the plane that contains $\vec{a}$ and $\vec{b}$.
Note that this cross-product can only be defined in *three-dimensional vector spaces*. The resulting vector
$\vec{c}=\vec{a}\times \vec{b} $ will have as components $c_1 = a_2b_3-a_3b_2$, $c_2= a_3b_1 - a_1b_3$, and $c_3= a_1b_2 - a_2b_1$.
The vector product (or cross product) between two vectors $\vec{a}$ and $\vec{b}$ is given by
where $|\vec{a}|=\sqrt{ \vec{a}\cdot\vec{a} }$ (and likewise for $|\vec{b}|$) is the norm of the vector $\vec{a}$, $\theta$ is the angle between the two vectors, and $\hat{n}$ is a unit vector which is *perpendicular* to the plane that contains $\vec{a}$ and $\vec{b}$.
Note that this cross-product can only be defined in *three-dimensional vector spaces*. The resulting vector
$\vec{c}=\vec{a}\times \vec{b} $ will have as components $c_1 = a_2b_3-a_3b_2$, $c_2= a_3b_1 - a_1b_3$, and $c_3= a_1b_2 - a_2b_1$.
- A special vector is the **unit vector**, which has a norm of 1 *by definition*. A unit vector is often denoted with a hat, rather than an arrow ($\hat{i}$ instead of $\vec{i}$). To find the unit vector in the direction of an arbitrary vector $\vec{v}$, we divide by the norm:
