Scalars and Vectors

A scalar has only magnitude. A vector has both magnitude and direction. This split organizes much of physics.

Concept

Scalar Examples

Mass, temperature, time, energy, work, density, speed, electric potential.

Vector Examples

Displacement, velocity, acceleration, force, momentum, electric field, torque.

Why Distinguish?

Vectors cannot be added like scalars: $5\,\text{N}$ east + $5\,\text{N}$ north is NOT $10\,\text{N}$. It is $5\sqrt{2}\,\text{N}$ at 45° north of east.

Representing a Vector

Arrow Notation

An arrow drawn from tail to head; length proportional to magnitude; direction along arrow.

Symbolic Notation

• $\vec{A}$ or $\mathbf{A}$ — boldface or arrow.
• Magnitude: $|\vec{A}|$ or simply $A$.
• Unit vectors: $\hat{i}, \hat{j}, \hat{k}$ along $x, y, z$.

Component Form

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

with magnitude $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$.

Types of Vectors

A unit vector in the direction of $\vec{A}$ is $\hat{A} = \frac{\vec{A}}{|\vec{A}|}.$

Worked Example

Q: A vector $\vec{A} = 3\hat{i} + 4\hat{j}$ m. Find (a) its magnitude, (b) the unit vector along $\vec{A}$, (c) the angle it makes with the $x$-axis.

Solution: (a) $|\vec{A}| = \sqrt{9 + 16} = 5$ m.

(b) $\hat{A} = (3/5)\hat{i} + (4/5)\hat{j} = 0.6\hat{i} + 0.8\hat{j}$.

(c) $\tan\theta = A_y/A_x = 4/3 \Rightarrow \theta = \arctan(4/3) \approx 53.1°$.

Common Confusions

• "Speed is a vector with direction." — No, speed is the magnitude of velocity. Velocity is the vector.
• "Negative quantities are vectors." — Not necessarily. Temperature can be negative but is still a scalar.
• "Equal vectors must be at the same location." — Vectors don't care where they are drawn; only magnitude and direction matter.

Key Takeaways

• Scalars: magnitude only. Vectors: magnitude + direction.
• Vectors are represented by arrows or components.
• Use $\hat{i}, \hat{j}, \hat{k}$ for Cartesian unit vectors.
• Unit vector along $\vec{A}$: $\hat{A} = \vec{A}/|\vec{A}|$.
• Pay attention to types: equal, negative, null, parallel, coplanar.