Vector Operations (March 3, 2026)

Just like regular numbers, vectors can be added and subtracted. This came out a lot in physics as well.

We simply have

A+B=\left\langle (a_1+b_1),(a_2+b_2),\cdots,(a_n+b_n) \right\rangle

A-B=\left\langle (a_1-b_1),(a_2-b_2),\cdots,(a_n-b_n) \right\rangle

When multiplying a scalar,

c\overrightarrow{A}=\left\langle ca_1,ca_2,\cdots,ca_n \right\rangle

We can note that the additive inverse of a vector A is -A and additive identity is 0, its multiplicative inverse (scalar) being nonexistent and its multiplicative identity being 1.

Unlike regular numbers, however, vectors have dot products and cross products.

Dot product:

A \cdot B= A_1B_1 + A_2B_2 + A_3B_3

In summation notation,

A \cdot B = \sum_{i=1}^{3} A_i B_i

In terms of the angles between the vectors,

A \cdot B = |A|\,|B| \cos\theta

Cross product:

A \times B = \begin{vmatrix} i & j & k \\ A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \end{vmatrix}

= i \begin{vmatrix} A_2 & A_3 \\ B_2 & B_3 \end{vmatrix} – j \begin{vmatrix} A_1 & A_3 \\ B_1 & B_3 \end{vmatrix} + k \begin{vmatrix} A_1 & A_2 \\ B_1 & B_2 \end{vmatrix}

= i(A_2B_3 – A_3B_2) – j(A_1B_3 – A_3B_1) + k(A_1B_2 – A_2B_1)

= \langle A_2B_3 – A_3B_2,\; A_3B_1 – A_1B_3,\; A_1B_2 – A_2B_1 \rangle

I knew about these from physics already, but I newly learned about the Levi-Civita symbol,

\varepsilon_{ijk} = \begin{cases} +1 & \text{if } (i,j,k) \text{ is an even permutation of } (1,2,3), \\ -1 & \text{if } (i,j,k) \text{ is an odd permutation of } (1,2,3), \\ 0 & \text{if any two indices are equal.} \end{cases}

(A \times B)_i = \sum_{j=1}^{3} \sum_{k=1}^{3} \varepsilon_{ijk} A_j B_k

We use this for a more compact notation that simplifies vector calculations.

The magnitude of the cross multiplication can be expressed as

|A \times B| = |A|\,|B| \sin\theta

I will explain the usages of these products in the next section.

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