Vector Concept

Vectors Have Magnitude and Direction

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

When the load is off-center, the cables no longer carry equal tension. Their directions determine the tension each cable needs to hold the load.

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 $27^\circ\text{C}$	temperature	scalar
A car travels a distance of $80 \text{ km}$	distance	scalar
A car is displaced $80 \text{ km}$ north	displacement	vector
A student pushes a table with $30 \text{ N}$ to the right	force	vector
Wind moves at $6 \text{ m s}^{-1}$ 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.

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, the magnitude of a force directed to the right can be written as $\lvert\vec{F}\rvert = 20 \text{ N}$ . The symbol $\vec{F}$ denotes the force vector, $20 \text{ N}$ is its magnitude, and "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 $B$ .

$\lvert\overrightarrow{AB}\rvert$ is the length of the vector from $A$ to $B$ .

The absolute value bars around a vector denote its magnitude. Direction is not part of that value.

Vector Equality Depends on Magnitude and Direction

Two vectors are equal if they have the same magnitude and the same direction. Their locations may be different.

Two forces of $15 \text{ N}$ directed to the right have equal vector values, even when the arrows are drawn at different locations. The point where a force acts affects the torque on a rigid body. Moving that point can change the torque even when the magnitude and direction stay 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.

Distance and Displacement Measure Different Quantities

A person 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 the vector from the starting position directly to the final position, so it differs from the total distance traveled.

\begin{aligned} \lvert\Delta\vec{r}\rvert &= \sqrt{4^2 + 3^2} \\ &= 5 \text{ m} \end{aligned}

The distance is $7 \text{ m}$ . The displacement has magnitude $5 \text{ m}$ and points about $36.9^\circ$ north of east.