For AI agents: use /llms.txt for the Nakafa content index.
Imagine telling a friend, "walk ." Your friend will probably ask, "which way?" Forward, right, or up the stairs?
In physics, some quantities cannot be explained by a number alone. They need both magnitude and direction. This kind of quantity is called a vector.
Examples include a displacement of east, a force of upward, or a velocity of to the right. If the direction is missing, the physical information is incomplete.
In this model, a small load moves along a bridge deck. Two cables pull the load toward the towers. Each cable pull is a vector because it has a force magnitude and a pulling direction. Move the load and notice how the arrow length and direction change together.
Here is the important point: when the load is not exactly in the middle, the two cables no longer pull with the same tension. The cable directions change, so the required tension magnitudes also change to keep the load supported. In physics, direction is not decoration. Direction changes the calculation.
Not every quantity needs direction. A scalar is described well enough by a value and a unit. A vector needs a value, a unit, and a direction.
| Situation | Quantity | Type |
|---|---|---|
| Water with mass | mass | scalar |
| Room temperature | temperature | scalar |
| A car travels a distance of | distance | scalar |
| A car is displaced north | displacement | vector |
| A student pushes a table with to the right | force | vector |
| Wind moves at west | velocity | vector |
A quick way to decide is this: if the question "which way?" matters for understanding the quantity, you are probably looking at a vector.
Vectors are often drawn as arrows. The tail shows where the vector starts, the head shows the direction, and the length represents the vector magnitude.
For example, a force can be written as to the right. The symbol shows that force is a vector. The number is its magnitude. The phrase "to the right" is its direction.
A vector can also be written from one point to another, such as . This means the vector starts at point and ends at point .
The absolute value bars around a vector mean we are talking only about its magnitude, not its direction.
Two vectors are equal if they have the same magnitude and the same direction. Their locations may be different.
Imagine two students pushing two different tables. Both push with a force of to the right. Even though the tables are in different places, the force vectors can be considered equal because their magnitude and direction are the same.
By contrast, a force of to the right and a force of to the left are not equal vectors. Their magnitudes are equal, but their directions are opposite.
A student walks east, then north. The distance traveled is the total path length:
Displacement is different. Displacement is the vector from the starting position directly to the final position.
So, the distance is , while the displacement magnitude is directed diagonally northeast. That is why vectors help: they tell us not only "how much", but also "which way".
