Let's delve into the main vector operations:

1. Vector Addition

Standard Definition: Given two vectors $\mathbf{A}$ and $\mathbf{B}$, their sum, $\mathbf{A} + \mathbf{B}$, is a third vector that is obtained by placing the tail of $\mathbf{B}$ at the head of $\mathbf{A}$ and then drawing a vector from the tail of $\mathbf{A}$ to the head of $\mathbf{B}$.

ELI5 with Real-life Example:
Imagine you're playing a video game where your character can walk in straight paths. If your character walks 5 steps forward and then 3 steps to the right, the total walk can be represented as the diagonal of a rectangle where one side is 5 steps and the other is 3 steps. This diagonal is the "sum" of your two walks. In aviation, planes adjust their paths for wind. If a plane wants to go north, but there's a wind pushing it east, its actual path will be a mix of north and east, found using vector addition.

2. Vector Subtraction

Standard Definition: To subtract vector $\mathbf{B}$ from $\mathbf{A}$, we add the negative of $\mathbf{B}$ to $\mathbf{A}$. So, $\mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B})$.

ELI5 with Real-life Example:
Think of it as a treasure hunt. If you first walk 10 steps to the north (vector $\mathbf{A}$) and then decide to retrace 6 steps back to the south (vector $\mathbf{B}$), the difference (or your net movement) is 4 steps to the north. In essence, you "subtracted" the southward walk from the northward walk.

3. Scalar Multiplication

Standard Definition: When a vector $\mathbf{A}$ is multiplied by a scalar (a regular number) $k$, the result is a new vector that has its magnitude scaled by $k$ and points in the same (or opposite, if $k < 0$) direction as $\mathbf{A}$.

ELI5 with Real-life Example:
Consider a toy car that can move in a straight line. If you push it gently, it might move a short distance (let's say 2 meters). But if you push it three times harder, it'll move three times that distance (6 meters). This is like multiplying the car's movement vector by the scalar "3".

These operations are foundational in vector calculus and have applications in physics, engineering, computer graphics, and many other fields.