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Vectors.md

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+#Math #Algebra
+
+# Defining Vectors
+
+Vectors are a list of components. They can be expressed in ij notation by:
+
+$$
+\mathbf a = 2i + 3j -4k
+$$
+
+or
+
+$$
+\vec a = 2i + 3j -4k
+$$
+
+You can also express a vector as a matrix:
+
+$$
+\vec a = \begin {bmatrix}
+2 \\
+3 \\
+-4 \\
+\end {bmatrix} \\
+\vec a = \begin {bmatrix} 2 & 3 & -4 \end {bmatrix}
+$$
+
+# Adding and Subtracting Vectors
+
+To add vectors, add their corresponding components. For example:
+
+$$
+\vec a = 4i + 7j - 9k \\
+\vec b = 3i - 5j - 8k \\
+\vec a + \vec b = 7i + 2j - 17k
+$$
+
+Subtracting vectors works in a similar fashion:
+
+$$
+\vec a - \vec b = i + 12j - k
+$$
+
+Here are the formulas:
+
+$$
+\vec a + \vec b = a_i+b_i \\
+\vec a - \vec b = a_i-b_i
+
+$$
+
+Here’s a graph visualizing the addition and subtraction of vectors: [https://www.desmos.com/calculator/gavjpwhnuo](https://www.desmos.com/calculator/gavjpwhnuo)
+
+# Multiplication by Scalar
+
+To multiply a vector by a scalar (regular number), just multiply all the components by that number:
+
+$$
+m\vec a = ma_i
+$$
+
+# Multiplication by Another Vector: Dot Product
+
+There are two different ways to multiply a vector by another vector. The first way is a dot product. Here is the algebraic definition, where n is the length of the two vectors:
+
+$$
+\vec a \cdot \vec b = \sum _{i = 0}^n a_ib_i
+$$
+
+With two two dimensional vectors, we can also provide a geometric definition, where $||\vec a||$ is the magnitude of $\vec a$, and $\theta$ is the angle between the vectors:
+
+$$
+\vec a \cdot \vec b = ||a|| \: ||b|| \: \cos \theta
+$$
+
+As you can see, the dot product returns a single value, or scalar. From the geometric definition, you can see that it describes how much one vector “aligns” to the other.
+
+## Proving that the Definitions are the Same
+
+Let $\vec a$ have a magnitude of $m$ and an angle of $p$, let $\vec b$ have a magnitude of $n$ and an angle of $q$.
+
+$$
+\vec a \cdot \vec b \\
+= m\cos p \: n \cos q + m\sin p \: n \sin q \\
+= mn(\cos p \: cos q + \sin p \: \sin q) \\
+= mn\cos(p-q)
+$$
+
+Using the algebraic definition, we can get the geometric definition as shown above.
+
+# Cross Product
+
+Let $n$ be a unit vector perpendicular to $\vec a$ and $\vec b$, and $\theta$ be the angle between them. The cross product is:
+
+$$
+\vec a \times \vec b = ||\vec a|| \: ||\vec b|| \: \sin \theta \: n
+$$