From 191d3f046874d7892d759507682d518f46e7373b Mon Sep 17 00:00:00 2001
From: Maciej Topyla <m.m.topyla@student.tudelft.nl>
Date: Sun, 4 Sep 2022 11:41:03 +0000
Subject: [PATCH] \hspace fix
---
src/3_vector_spaces.md | 17 +++++++++--------
1 file changed, 9 insertions(+), 8 deletions(-)
diff --git a/src/3_vector_spaces.md b/src/3_vector_spaces.md
index e380fd2..2e13ec3 100644
--- a/src/3_vector_spaces.md
+++ b/src/3_vector_spaces.md
@@ -90,16 +90,17 @@ 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
- $$1. \vspace{5pt} (\lambda \mu) \vec{a} = \lambda (\mu \vec{a}) = \mu (\lambda \vec{a})$$
- $$2. \vspace{5pt} \lambda (\vec{a} + \vec{b}) = \lambda \vec{a} + \lambda \vec{b}$$
- $$3. \vspace{5pt} (\lambda + \mu)\vec{a} = \lambda \vec{a} +\mu \vec{a}$$
+ $$1. \hspace{5pt} (\lambda \mu) \vec{a} = \lambda (\mu \vec{a}) = \mu (\lambda \vec{a})$$
+ $$2. \hspace{5pt} \lambda (\vec{a} + \vec{b}) = \lambda \vec{a} + \lambda \vec{b}$$
+ $$3. \hspace{5pt} (\lambda + \mu)\vec{a} = \lambda \vec{a} +\mu \vec{a}$$
+
+### Vector products
-!!! 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.
+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 $$ \vec{a}\cdot \vec{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \, .$$
+!!! info "Scalar product of vectors"
+ The scalar product 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.
--
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