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Removed bullet points to support array/matrix like syntax in Latex
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.obsidian/appearance.json
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Mathematics/Linear Algebra/Vectors.md
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OdyAsh committed Nov 4, 2023
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2 changes: 1 addition & 1 deletion .obsidian/appearance.json
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3 changes: 1 addition & 2 deletions .obsidian/workspace.json
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31 changes: 20 additions & 11 deletions Mathematics/Linear Algebra/Vectors.md
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Expand Up @@ -5,23 +5,29 @@ sources:

# Vector Definition

Basic vector characteristics:
* It is something that has a magnitude and a direction.
* Intuitive example ([source](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/vector-introduction-linear-algebra)):
![](Attachments%20-%20Vectors/Pasted%20image%2020231104130417.png)
* It has the following notation:
<br><img src="https://latex.codecogs.com/svg.image?\vec{v}=\left(5,0\right)=\left[\begin{array}{l}{5}\cr{0}\end{array}\right]" height=auto>
* $5$ and $0$ are called ***components*** of a vector, while $(5,0)$ is called a ***2-tuple*** in a 2-D real coordinate space $\left(\mathbb{R}^2\right)$ ([source](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/real-coordinate-spaces)).
* Side note 1: $\mathbb{R}^2$ is also called a ***set***, such that $\vec{v}\in\mathbb{R}^2$ is called " vector $\vec{v}$ belongs in the set $\mathbb{R}^2$ "
* Side note 2: it can also be represented using unit vectors $i$ and $j$ , as explained in the [Unit vector](#Unit%20vector) section.
It is something that has a magnitude and a direction.

Intuitive example ([source](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/vector-introduction-linear-algebra)):
![](Attachments%20-%20Vectors/Pasted%20image%2020231104130417.png)

# Vector Notation

It has notation like the following:
$$\vec{v}=\left(5,0\right)=\left[\begin{array}{l}{5}\cr{0}\end{array}\right]=5i+0j$$
Such that:
* $5$ and $0$ are called ***components*** of a vector, while $(5,0)$ is called a ***2-tuple*** in a 2-D real coordinate space $\left(\mathbb{R}^2\right)$ ([source](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/real-coordinate-spaces)).
* Side note 1: $\mathbb{R}^2$ is also called a ***set***, such that $\vec{v}\in\mathbb{R}^2$ is called " vector $\vec{v}$ belongs in the set $\mathbb{R}^2$ "
* It can also be represented using unit vectors $i$ and $j$ , as explained in the [Unit vector](#Unit%20Vector) section.
* It can be [added](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/adding-vectors) and [multiplied (by scalers)](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/v/multiplying-vector-by-scalar)


## Vector Addition

Addition visualization ([interactive source](https://sciencepickleapps.com/VisuallyAddingVectorsV1-0-0/), from [this](https://sciencepickle.com/earth-systems/vectors-and-forces/adding-vectors/)):
![vector-addition](Attachments%20-%20Vectors/vector-addition.mp4)

Example: $\vec{v}=\vec{red}+\vec{green}+\vec{blue}=\left[\begin{array}{c}{-9}\cr{3}\end{array}\right]+\left[\begin{array}{c}{2}\cr{3}\end{array}\right]+\left[\begin{array}{c}{2}\cr{0.2}\end{array}\right]=\left[\begin{array}{c}{-5}\cr{6.2}\end{array}\right]$
Example:
$$\vec{v}=\vec{red}+\vec{green}+\vec{blue}=\left[\begin{array}{c}{-9}\cr{3}\end{array}\right]+\left[\begin{array}{c}{2}\cr{3}\end{array}\right]+\left[\begin{array}{c}{2}\cr{0.2}\end{array}\right]=\left[\begin{array}{c}{-5}\cr{6.2}\end{array}\right]$$

## Vector Multiplication

Expand All @@ -33,11 +39,14 @@ Formula of a scalar-multiplied vector's magnitude: $||c\cdot \vec v||=|c|\cdot |
To get the reverse direction of a vector, add 180 to its direction.

Example ([source](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/vectors/e/scaling_vectors)):

![](Attachments%20-%20Vectors/Pasted%20image%2020231104110053.png)
Answer:

![](Attachments%20-%20Vectors/Pasted%20image%2020231104112518.png)

![](Attachments%20-%20Vectors/Pasted%20image%2020231104112617.png)


## Unit vector
## Unit Vector

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