diff --git a/src/3_vector_spaces.md b/src/3_vector_spaces.md index 582c9f7c046d927d7e8ccb21e12ea24d17f696e3..63a62b495309163d381d6c21ddc7062205f491a6 100644 --- a/src/3_vector_spaces.md +++ b/src/3_vector_spaces.md @@ -90,28 +90,26 @@ You might be already familiar with the concept of performing a number of various !!! info "Scalar multiplication" I can multiply a vector by a scalar number (either real or complex) to produce another vector, $$\vec{c} = \lambda \vec{a}$$. Addition and scalar multiplication of vectors are both *associative* and *distributive*, so the following relations hold - 1. $(\lambda \mu) \vec{a} = \lambda (\mu \vec{a}) = \mu (\lambda \vec{a})$ 2. $\lambda (\vec{a} + \vec{b}) = \lambda \vec{a} + \lambda \vec{b}$ 3. $(\lambda + \mu)\vec{a} = \lambda \vec{a} +\mu \vec{a}$ -- **Vector product**: 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. - -- 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 **vector product** (or cross product) between two vectors $\vec{a}$ and $\vec{b}$ is given by -$$ -\vec{a}\times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n} \, , -$$ -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 $c_3= a_1b_2 - a_2b_1$. +!!! info "Vector product" + 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. + +!!! info "Cross product" + "The vector product (or cross product) between two vectors $\vec{a}$ and $\vec{b}$ is given by + $$ \vec{a}\times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n}$$ + 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: $$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$$