Chapter 1 Notes: Units, Physical Quantities, and Vectors

Learning Outcomes

• The four steps you can use to solve any physics problem:

  • Identify the relevant concepts, target variables, and known quantities, as stated or implied in the problem.

  • Set Up the problem: choose the equations you’ll use to solve the problem, and draw a sketch of the situation.

  • Execute the solution: do the mathematics.

  • Evaluate your answer: compare with your estimates and reconsider if there’s a discrepancy.

• Three fundamental quantities of physics and the units physicists use to measure them.

• How to work with units and significant figures in calculations.

• How to add and subtract vectors graphically, and using vector components.

• Two ways to multiply vectors: the scalar (dot) product and the vector (cross) product.

The Nature of Physics

• Physics is an experimental science in which physicists seek patterns that relate the phenomena of nature.

• The patterns are called physical theories.

• A very well established or widely used theory is called a physical law or principle.

Solving Problems in Physics

• The Problem-Solving Strategy used throughout this book follows four steps:

  • Identify the relevant concepts, target variables, and known quantities, as stated or implied in the problem.

  • Set up the problem: choose the equations you’ll use and draw a sketch.

  • Execute the solution: do the math.

  • Evaluate your answer: compare with estimates and reconsider if there’s a discrepancy.

Idealized Models

• To simplify analysis, we use idealized models.

• Example: a baseball in flight is analyzed using an idealized model for tractability.

Standards and Units

• Length, time, and mass are three fundamental quantities of physics.

• The International System (SI) is the most widely used system of units.

• In SI units:

  • Length is measured in meters: $L = ext{m}$

  • Time in seconds: $T = ext{s}$

  • Mass in kilograms: $M = ext{kg}$

Unit Prefixes

• Prefixes create larger and smaller units for fundamental quantities. Examples:

  • Kilometer: $1~ ext{km} = 10^{3}~ ext{m}$

  • Meter to millimeter: $1~ ext{m} = 10^{3}~ ext{mm}$

  • Micrometer: $1~ ext{μm} = 10^{-6}~ ext{m}$

  • Nanosecond: $1~ ext{ns} = 10^{-9}~ ext{s}$

  • Gram to kilogram: $1~ ext{g} = 10^{-3}~ ext{kg}$

• (Other prefixes include milli, micro, nano, kilo, mega, giga, etc.)

Unit Consistency and Conversions

• An equation must be dimensionally consistent: terms added or equated must have the same units (you’re always converting apples to apples).

• Always carry units through calculations.

• Convert to standard units as necessary by forming a ratio of the same physical quantity in two different units and using it as a multiplier.

• Example: seconds in 3 minutes:
  $3~ ext{min} imes rac{60~ ext{s}}{1~ ext{min}} = 180~ ext{s}$

Uncertainty and Significant Figures

• The uncertainty of a measured quantity is indicated by its number of significant figures.

• For multiplication and division:

  • The result can have no more significant figures than the smallest number of significant figures in the factors.

• For addition and subtraction:

  • The number of significant figures is determined by the term with the fewest digits to the right of the decimal point.

• Even small percent errors can have large consequences (illustrated by the train mishap reference).

• Video Tutor Solution: Example 1.3 (illustrative support).

Vectors and Scalars

• A scalar quantity can be described by a single number.

• A vector quantity has both a magnitude and a direction in space.

• In this book, a vector quantity is represented in boldface italic with an arrow over it, e.g. $\vec{A}$.

• The magnitude of a vector is denoted as $|\vec{A}|$.

• A vector can be written in component form as $\vec{A} = A<em>x\hat{\mathbf{i}} + A</em>y\hat{\mathbf{j}} + A_z\hat{\mathbf{k}}$ (3D). In 2D, omit the z-term.

Drawing Vectors

• A vector is drawn as a line with an arrowhead at its tip.

• The length of the line represents magnitude; the direction of the line represents direction.

Adding Two Vectors Graphically (1 of 3)

• Head-to-tail method: place the tail of B at the head of A; the resultant vector C goes from the tail of A to the head of B.

• Commutativity: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$

Adding Two Vectors Graphically (2 of 3)

• Tail-to-tail method: construct a parallelogram; the diagonal from the common tail gives the sum: $\vec{C} = \vec{A} + \vec{B}$

Adding Two Vectors Graphically (3 of 3)

• If A and B are parallel, the magnitude of the sum is simply: $|\vec{C}| = |\vec{A} + \vec{B}| = |\vec{A}| + |\vec{B}|$

• If A and B are antiparallel, $|\vec{C}| = |\,|\vec{A}| - |\vec{B}|\,|$

• In all cases, the vector sum is given by the head-to-tail result: $\vec{C} = \vec{A} + \vec{B}$

Adding More Than Two Vectors Graphically (1–3)

• To add several vectors, use the head-to-tail method. The vectors can be added in any order.

Subtracting Vectors

• Subtracting B from A: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

• Graphically, with A and B head-to-tail, A - B is the vector from the tail of A to the head of -B.

• Alternative: with A and B parallel, A - B is the vector from the tail of A to the tail of B.

Multiplying a Vector by a Scalar

• If c is a scalar, the product $c\vec{A}$ has magnitude $|c|\,|\vec{A}|$.

• Sign of the scalar affects direction: a positive scalar preserves direction, a negative scalar reverses it.

Addition of Two Vectors at Right Angles

• For perpendicular vectors, add graphically first, then use trigonometry to find the magnitude and direction of the sum.

• Example (from the material): a cross-country skier ends up $2.24\ \text{km}$ from the starting point, in a direction of $63.4^{\circ}$ east of north.

Components of a Vector

• Any vector can be represented by its components in a given coordinate system.

• In 2D, $\vec{A} = A<em>x\hat{\mathbf{i}} + A</em>y\hat{\mathbf{j}}$; in 3D, add $+ A_z\hat{\mathbf{k}}$

Positive and Negative Components

• Components can be positive or negative depending on direction relative to the axes.

Finding Components

• Given magnitude and direction, components are:
  $A<em>x = A\cos\theta, \quad A</em>y = A\sin\theta$
  (for angle \theta measured from the +x-axis).

• The text notes that video examples illustrate these procedures (Example 1.6).

Calculations Using Components

• Magnitude and direction from components:
  $|\vec{A}| = \sqrt{A<em>x^2 + A</em>y^2 + A<em>z^2}$ $\theta = \tan^{-1}\left( \frac{A</em>y}{A_x} \right)$

• The components of a sum relate component-wise:
  $\vec{R} = \vec{A} + \vec{B} + \vec{C}$
  $R<em>x = A</em>x + B<em>x + C</em>x, \quad R<em>y = A</em>y + B<em>y + C</em>y, \quad R<em>z = A</em>z + B<em>z + C</em>z$

Unit Vectors

• A unit vector has magnitude 1 and no units: $|\,\hat{\mathbf{u}}| = 1$.

• Standard unit vectors: $\hat{\mathbf{i}}$ along +x, $\hat{\mathbf{j}}$ along +y, and $\hat{\mathbf{k}}$ along +z.

• Any vector can be expressed as a linear combination of unit vectors: $\vec{A} = A<em>x\hat{\mathbf{i}} + A</em>y\hat{\mathbf{j}} + A_z\hat{\mathbf{k}}.$

The Scalar Product (Dot Product) (1 of 2)

• When placed tail to tail, the scalar product is:
  $\vec{A} \cdot \vec{B} = AB\cos\phi$
  where $\phi$ is the angle between A and B.

• In another form, the dot product equals magnitude times the component of one vector in the direction of the other:
  $\vec{A} \cdot \vec{B} = |\vec{A}|\,|\vec{B}|\cos\phi = A\,(B\cos\phi)$

• Alternatively, $\vec{A} \cdot \vec{B} = B\,(A\cos\phi)$ depending on which is treated as the base.

The Scalar Product (Dot Product) (2 of 2)

• The scalar product can be positive, negative, or zero depending on the angle: $\vec{A} \cdot \vec{B} \in \mathbb{R}$ with sign determined by \phi.

Calculating a Scalar Product Using Components

• In components: $\vec{A} \cdot \vec{B} = A<em>x B</em>x + A<em>y B</em>y + A<em>z B</em>z.$

Finding an Angle Using the Scalar Product

• To find the angle between two vectors from their components, use:
  $\cos\phi = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}|\,|\vec{B}|}.$

• Then $\phi = \cos^{-1}\left( \frac{\vec{A} \cdot \vec{B}}{|\vec{A}|\,|\vec{B}|} \right).$

The Vector Product (Cross Product) (1 of 2)

• The vector product of two vectors A and B is defined as:
  $\vec{C} = \vec{A} \times \vec{B}$

• The direction of $\vec{C}$ is given by the right-hand rule.

• The cross product is anticommutative: $\vec{A} \times \vec{B} = -\big( \vec{B} \times \vec{A} \big).$

The Vector Product (Cross Product) (2 of 2)

• Magnitude of the vector product is:
  $|\vec{C}| = |\vec{A}|\,|\vec{B}|\sin\phi,$
  where $\phi$ is the angle between $\vec{A}$ and $\vec{B}$.

• The cross product is perpendicular to the plane containing A and B.

Calculating the Vector Product

• Use the magnitude formula: $|\vec{C}| = |\vec{A}|\,|\vec{B}|\sin\phi$

• Determine the direction using the right-hand rule.

• The material references Example 1.11 for practice and a Video Tutor Solution for Example 1.11.

Summary of Key Relationships

• Dot product: $\vec{A} \cdot \vec{B} = |\vec{A}|\,|\vec{B}|\cos\phi = A<em>x B</em>x + A<em>y B</em>y + A<em>z B</em>z.$

• Cross product: $\vec{C} = \vec{A} \times \vec{B}, \quad |\vec{C}| = |\vec{A}|\,|\vec{B}|\sin\phi, \quad \text{direction given by the right-hand rule}, \vec{A} \times \vec{B} = - (\vec{B} \times \vec{A}).$

• Angle from dot product: $\cos\phi = \dfrac{\vec{A} \cdot \vec{B}}{|\vec{A}|\,|\vec{B}|}.$

Connections to Foundational Concepts

• Vectors vs Scalars: vectors have both magnitude and direction; scalars have only magnitude.

• Unit vectors provide a convenient basis to express any vector in a coordinate system.

• Component form allows straightforward arithmetic for sums and magnitudes.

• The dot product relates to projections of one vector onto another.

• The cross product relates to area of parallelograms spanned by the vectors and to a vector perpendicular to that plane.

Practical and Philosophical Implications

• Using idealized models helps simplify complex real-world systems while preserving essential physics.

• Dimensional analysis (unit consistency) enforces physical meaning and prevents nonsensical results.

• Understanding components and vector operations builds intuition for more advanced topics (e.g., dynamics, electromagnetism).

Additional Notes and Examples References

• Perpendicular vectors: $|\vec{A} + \vec{B}| = \sqrt{A^2 + B^2}$ when $\vec{A} \perp \vec{B}$.

• For right-angle vectors, angle tracking with arctangent relationships and trigonometry is essential for direction.

• When combining multiple vectors, the order of addition does not matter (associativity and commutativity of addition).

Quick Reference Formulas

Scalar Product (Dot Product)

This gives a single number (a scalar) from two vectors. It shows how much one vector points in the direction of another. It's used in things like calculating work.

• Using components: $\vec{A} \cdot \vec{B} = Ax Bx + Ay By + Az Bz$

• Using magnitudes and angle: $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos\phi$ (where $\phi$ is the angle between $\vec{A}$ and $\vec{B}$)

Vector Product (Cross Product)

This gives a new vector that is at right angles to both original vectors. Its length relates to the area of the parallelogram made by the vectors, and its direction is found using the right-hand rule. It's used in things like torque.

• The new vector is $\vec{C} = \vec{A} \times \vec{B}$.

• Magnitude (length) of the new vector: $|\vec{C}| = |\vec{A}| |\vec{B}| \sin\phi$ (where $\phi$ is the angle between $\vec{A}$ and $\vec{B}$)

• Direction: Found using the right-hand rule.

• Order matters: $\vec{A} \times \vec{B} = - (\vec{B} \times \vec{A})$

Vector Magnitude from Components

This formula finds the total length (or magnitude) of a vector from its separate x, y, and z parts.

• For a vector $\vec{A}$ with parts $(Ax, Ay, Az)$: $|\vec{A}| = \sqrt{Ax^2 + Ay^2 + Az^2}$

Vector Components from Magnitude and Direction

This shows how to break down a vector into its horizontal (x) and vertical (y) parts if you know its total length and angle.

• For a vector $\vec{A}$ with length $A$ and angle $\theta$ (from the positive x-axis):

  • X-component: $Ax = A\cos\theta$

  • Y-component: $Ay = A\sin\theta$

Angle from Components

This formula helps you find the direction (angle) of a vector when you know its x and y parts.

• The angle $\theta$ (from the positive x-axis) of a vector $\vec{A}$ with parts $(Ax, Ay)$: $\theta = \tan^{-1}\left(\dfrac{Ay}{Ax}\right)$

• Important: Remember to check the signs of Ax and Ay to make sure your angle is in the correct quadrant (e.g., top-right, top-left, bottom-left, bottom-right).

Components of a Resultant Vector (Adding Vectors by Components)

To add several vectors together, you simply add their corresponding x-parts, y-parts, and z-parts separately.

• For a total vector $\vec{R} = \vec{A} + \vec{B} + \vec{C}$:

  • X-parts sum: $Rx = Ax + Bx + Cx$

  • Y-parts sum: $Ry = Ay + By + Cy$

  • Z-parts sum: $Rz = Az + Bz + Cz$

Sum of Two Vectors (Drawing Methods)

These are ways to add two vectors visually.

• Head-to-tail method: Put the start of the second vector ($\vec{B}$'s tail) at the end of the first vector ($\vec{A}$'s head). The sum ($\vec{A} + \vec{B}$) goes from the start of $\vec{A}$ to the end of $\vec{B}$.

• Parallelogram method: Put both vectors' starts (tails) at the same point. Draw a parallelogram using them. The sum ($\vec{A} + \vec{B}$) is the diagonal starting from their common tail.

Subtracting Vectors

Subtracting vector $\vec{B}$ from $\vec{A}$ is the same as adding $\vec{A}$ to the negative of $\vec{B}$ (which means $\vec{B}$ pointing in the opposite direction).

• $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

Scalar Multiple of a Vector

Multiplying a vector by an ordinary number (a scalar) changes its length and sometimes its direction.

• If $c$ is a scalar and $\vec{A}$ is a vector, the product $c\vec{A}$ has a length of $|c| |\vec{A}|$.

• If $c$ is positive, $c\vec{A}$ points in the same direction as $\vec{A}$.

• If $c$ is negative, $c\vec{A}$ points in the opposite direction