Saying "the temperature is 25 degrees" has no direction. One number is enough. But to say "the wind is blowing," you need not just its strength (how many meters per second) but also its direction (which way it's coming from) for the information to be complete. A quantity like this, that has both size and direction at once, is called a vector, and it's represented as an arrow. The length of the arrow shows its size, and the direction it points shows its direction.
Written in coordinates, a vector looks like u=(uₓ, uᵧ). This means "move uₓ along the x-axis and uᵧ along the y-axis." The way you add two vectors is interesting: connect the tail of v's arrow to the tip of u's arrow, and draw a new arrow from the very first starting point to the very last ending point — that new arrow is exactly u+v (the "triangle rule"). When you calculate with coordinates, all you do is add each component separately: (uₓ+vₓ, uᵧ+vᵧ).
Among the ways to multiply vectors together, the dot product (u·v = uₓvₓ+uᵧvᵧ) has a special meaning. The dot product tells you, with a single number, "how much these two vectors line up pointing the same direction." A large positive dot product means the two vectors point in similar directions; a negative one means they're closer to opposite directions. And the most important property of all — if the dot product is exactly 0, the two vectors must be perpendicular. You can confirm this directly from the dot product's other formula, u·v=|u||v|cos θ. When θ=90°, cos 90°=0, so the dot product has no choice but to be 0 too.
The property "dot product 0 means perpendicular" is extremely useful in practice. When you want to check whether two lines are perpendicular, you don't need a protractor — just compute the dot product of their two direction vectors and check whether it's 0. This same principle gets used in architecture and robotics too, to check whether a wall stands at exactly a right angle, or whether a robot arm moved through exactly the right angle.
Does the formula for a vector's magnitude, |u| = √(uₓ²+uᵧ²), look familiar? It has the exact same shape as the distance formula you learned with coordinates. In fact, a vector can be thought of as "an arrow from the origin to some point," so a vector's magnitude is exactly the same as the distance to that point. Coordinates, distance, and vectors might seem like different names for different things, but they're really one connected idea.
When studying this with kids, it helps to have them draw an actual travel route on a map, like "3km east, then 4km north," as arrows. Have them connect the two legs of the trip and calculate where they actually end up, and it becomes natural to understand why vector addition is defined the way it is. On our activity page, you can type in the coordinates of two vectors yourself to see how addition works, and change the angle between two vectors to find the exact moment their dot product hits 0 (perpendicular).