Defining Vectors

Vectors are a list of components. They can be expressed in ij notation by:


\mathbf a = 2i + 3j -4k


\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

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

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