Vector Concept

A Number Alone Is Not Enough

Imagine telling a friend, "walk $5 \text{ m}$ ." 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.

\text{vector} = \text{magnitude} + \text{direction}

Examples include a displacement of $5 \text{ m}$ east, a force of $20 \text{ N}$ upward, or a velocity of $12 \text{ m s}^{-1}$ to the right. If the direction is missing, the physical information is incomplete.

Seeing Vectors on a Bridge

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.

Cable Tension as a Vector

Move the load along the bridge. The arrows show the cable tension forces acting on the load.

Move the load position

0 \text{ m}

Magnitude: shown by the arrow length
Direction: shown by the arrow tilt
Resultant: the combined effect of several vectors

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.

Vectors and Scalars

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 $2 \text{ kg}$	mass	scalar
Room temperature $27^\circ\text{C}$

A quick way to decide is this: if the question "which way?" matters for understanding the quantity, you are probably looking at a vector.

Reading a Vector Arrow

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 $\vec{F} = 20 \text{ N}$ to the right. The symbol $\vec{F}$ 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 $\overrightarrow{AB}$ . This means the vector starts at point $A$ and ends at point $B$ .

\lvert\overrightarrow{AB}\rvert = \text{the vector length from } A \text{ to } B

The absolute value bars around a vector mean we are talking only about its magnitude, not its direction.

Equal Vectors Do Not Need the Same Place

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 $15 \text{ N}$ 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 $15 \text{ N}$ to the right and a force of $15 \text{ N}$ to the left are not equal vectors. Their magnitudes are equal, but their directions are opposite.

A Wind Arrow Needs Both Clues

A student walks $4 \text{ m}$ east, then $3 \text{ m}$ north. The distance traveled is the total path length:

4 \text{ m} + 3 \text{ m} = 7 \text{ m}

Displacement is different. Displacement is the vector from the starting position directly to the final position.

\begin{aligned} s &= \sqrt{4^2 + 3^2} \\ &= 5 \text{ m} \end{aligned}

So, the distance is $7 \text{ m}$ , while the displacement magnitude is $5 \text{ m}$ directed diagonally northeast. That is why vectors help: they tell us not only "how much", but also "which way".