Vectors and Scalars: Foundations and Operations in Physics

3.2 Vectors and Scalars

Definition and Distinction

Physical quantities in physics are classified as either scalars or vectors based on whether they possess direction in addition to magnitude.

• Scalar: A quantity with only magnitude (numerical value with units), no direction.

• Vector: A quantity with both magnitude and direction.

Examples

Scalar Quantities	Vector Quantities
Speed, Distance, Mass, Energy, Density, Power, Length, Area, Volume, Time, Temperature, Work	Displacement, Velocity, Weight, Acceleration, Force, Impulse, Pressure, Momentum, Gravity, Drag

Key Differences

Scalar	Vector
Has magnitude only	Has both magnitude and direction
Ordinary algebra applies	Vector algebra applies
Resultant of two scalars is scalar	Resultant can be scalar or vector
Examples: mass, time, energy	Examples: velocity, force

Representation of Vectors

• Vectors are represented by arrows.

• The length of the arrow indicates magnitude.

• The head of the arrow indicates direction.

• Vectors may be denoted in bold (e.g., a) or with an arrow overhead (e.g., $\vec{a}$).

3.3 Adding Vectors Geometrically

Graphical Addition

Vectors are added using the head-to-tail method:

• Place the tail of the second vector at the head of the first.

• The resultant vector ($\vec{c}$) is drawn from the tail of the first to the head of the last vector.

Vector Addition Laws

• Commutative Law: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$

• Associative Law: $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$

• Zero Vector: $\vec{b} + ( -\vec{b}) = 0$

• Subtraction: $\vec{d} = \vec{a} - \vec{b} = \vec{a} + ( -\vec{b})$

Component Addition

• Vectors can be resolved into components along the x, y, and z axes.

• Resultant components: $c_x = a_x + b_x$, $c_y = a_y + b_y$

Example

Practice A: 10 km North + 5 km West. Resultant $R$ is found using the Pythagorean theorem.

3.4 Components of Vectors

Resolution of Vectors

The component of a vector along an axis is its projection onto that axis. The process of finding components is called resolution of the vector.

• In 2D: $R_x = R \cos \theta$, $R_y = R \sin \theta$

• In 3D: Components along x, y, z axes

• Magnitude: $a = \sqrt{a_x^2 + a_y^2}$

• Direction: $\tan \theta = \frac{a_y}{a_x}$

Unit Vectors

• Unit vectors have magnitude 1 and indicate direction along axes: $\hat{i}$ (x), $\hat{j}$ (y), $\hat{k}$ (z)

• Any vector $\vec{A}$ can be written as $A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$

Example

A plane travels 215 km at 22° east of due north. Components are found using trigonometric functions.

3.6 Adding Vectors by Components

Component Method

• If $\vec{r} = \vec{a} + \vec{b}$, then $r_x = a_x + b_x$, $r_y = a_y + b_y$, $r_z = a_z + b_z$

• Vector subtraction: $d_x = a_x - b_x$, $d_y = a_y - b_y$, $d_z = a_z - b_z$

Example

Given $\vec{a} = 2\hat{i} + 2\hat{j} - 5\hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$, the unit vector in the direction of $\vec{a} + \vec{b}$ is $\frac{1}{\sqrt{29}}(4\hat{i} + 3\hat{j} - 2\hat{k})$.

3.7 Vectors and the Laws of Physics

Coordinate System Independence

• Physical laws and vector relations do not depend on the choice of coordinate system or origin.

• Rotating axes changes components but not the vector itself.

3.8 Multiplying Vectors

A. Multiplying a Vector by a Scalar

• Changes the magnitude, not the direction: $\vec{a} \times s = s\vec{a}$

B. Scalar (Dot) Product

• Defined as $\vec{a} \cdot \vec{b} = ab \cos \phi$

• Result is a scalar quantity

• Commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$

• If vectors are perpendicular, dot product is zero

Dot Product in Components

• $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$

C. Vector (Cross) Product

• Defined as $\vec{a} \times \vec{b} = ab \sin \phi \; \hat{n}$, where $\hat{n}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$

• Result is a vector quantity

• Not commutative: $\vec{a} \times \vec{b} = - (\vec{b} \times \vec{a})$

• If vectors are parallel, cross product is zero

Cross Product in Components

• $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$

Right-Hand Rule

• Direction of the cross product is given by the right-hand rule: point fingers in direction of $\vec{a}$, curl toward $\vec{b}$, thumb points in direction of $\vec{a} \times \vec{b}$.

Dot vs. Cross Product Comparison

Dot Product	Cross Product
Product of magnitudes and cosine of angle	Product of magnitudes and sine of angle
Result is scalar	Result is vector
Commutative	Not commutative
Zero if vectors are perpendicular	Zero if vectors are parallel

Example

Find the angle between $\vec{a} = 2\hat{i} + 3\hat{j} + \hat{k}$ and $\vec{b} = 5\hat{i} - 2\hat{j} + 3\hat{k}$ using the dot product formula.

Summary Table: Scalar, Vector, and Their Products

Scalar Quantity	Vector Quantity	Product (Vector Quantity)
mass	velocity	momentum
time	velocity	displacement
5 kg	10 m/s (eastwards)	50 kg·m/s (eastwards)
5 s	10 m/s (eastwards)	50 m (eastwards)

Problem Solving Checkpoints

• Angles measured counterclockwise are positive; clockwise are negative.

• Use trigonometric functions and check calculator settings.

• Ensure angle units are consistent.

Additional info:

These notes cover the foundational concepts of vectors and scalars, their operations, and their importance in physics, suitable for introductory college-level physics courses.