2.5 Scalar vs. Vector Quantities

In physics, all measurable quantities are classified as either scalar or vector. The fundamental difference between them lies in whether the quantity has a direction associated with it.

Scalar Quantities

A scalar quantity is a physical quantity that is fully described by its magnitude (numerical value with unit) alone. Scalars obey the rules of ordinary arithmetic.

Examples of Scalar Quantities

Vector Quantities

A vector quantity requires both magnitude and direction for a complete description. Vectors are represented graphically by arrows — the length indicates magnitude and the arrowhead indicates direction. Vector algebra (not ordinary arithmetic) is used to manipulate them.

Examples of Vector Quantities

Distinguishing Key Pairs

Example: If you walk 5 m east then 5 m west, your distance = 10 m but your displacement = 0 m.

Rectangular Components Of A Vector→

Representing Vectors in 2-D: Rectangular Components

Any 2-D vector $\vec{A}$ can be resolved into two perpendicular components along the x- and y-axes:

$A_x=A\cos \theta ,A_y=A\sin \theta$

where $A=|\vec{A}|$ is the magnitude and $\theta$ is the angle with the positive x-axis. In component form:

$\vec{A}=A_x\hat{i}+A_y\hat{j}$

The unit vectors $\hat{i}$ and $\hat{j}$ point along the positive x- and y-axes respectively, each with magnitude 1.

Worked Example: A force of 10 N acts at 30° above the positive x-axis. Find its components.

$F_x=10\cos 30°=10\times \frac{\sqrt{3}}{2}\approx 8.66\text{ N}$

$F_y=10\sin 30°=10\times \frac{1}{2}=5\text{ N}$

So $\vec{F}=8.66\hat{i}+5\hat{j}$ N.

Special Types of Vectors

In the FSc curriculum, several specific types of vectors are essential:

Unit Vector

A vector having a magnitude of unity (1) used to specify direction. Denoted by a hat symbol, e.g., $\hat{a}$. The standard unit vectors $\hat{i}$, $\hat{j}$, $\hat{k}$ point along the positive x, y, and z axes respectively.

Null Vector (Zero Vector)

A vector with zero magnitude and an arbitrary (undefined) direction. It results from operations such as $\vec{A}-\vec{A}=\vec{0}$.

Position Vector

A vector that describes the location of a point relative to the origin $O$. In 3-D:

$\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$

where $x$, $y$, $z$ are the Cartesian coordinates of the point.

Comparison Table