Chapter 2 Notes (Vectors)

2.1 Scalars and Vectors

• Why vectors matter in physics and engineering

  • Many physical quantities are vectors (e.g., displacement, velocity, force, electric and magnetic fields).
  • Scalar products (dot products) define scalars like energy; vector products (cross products) define vector quantities like torque and angular momentum.
  • Vectors are Euclidean quantities with geometric representations as arrows in 1D, 2D, or 3D. They can be added, subtracted, or multiplied.
  • Preface: in this course, vector algebra is used for mechanics and electricity & magnetism; graphical methods exist but analytical methods are preferred for problems.

• Vector representation and notation

  • In physics, boldface with an arrow above denotes a vector, while a plain letter denotes a scalar. Examples:
  • scalar: distance d = 2.0 km
  • vector: displacement
    $\vec{r} \;\text{or}\; \mathbf{r}$
  • A vector has two key attributes: magnitude (length) and direction. Magnitude is a positive scalar; direction is given by the arrow.
  • Displacement is a vector that represents a change in position; its magnitude is the distance traveled and its direction is the direction of the displacement.

• One-dimensional vector algebra (in a line)

  • A vector can be multiplied by a scalar: if a vector (\mathbf{v}) is multiplied by a scalar (\lambda>0), the resultant (\lambda \mathbf{v}) is parallel to (\mathbf{v}) with magnitude increased by (|\lambda|). If (\lambda<0), the direction is antiparallel.
  • Addition of parallel vectors along a line yields a magnitude equal to the sum of magnitudes; the resultant is parallel to the originals.
  • Subtraction corresponds to adding the antiparallel vector: if (\mathbf{b}= -\mathbf{a}), then ( \mathbf{a} - \mathbf{b} = 2\mathbf{a}).
  • Unit vector: a vector of magnitude 1 that points in a specified direction. Denoted with a hat, e.g., (\hat{\mathbf{e}}_x).
  • Any vector can be written as a magnitude times a unit direction: $\vec{v} = v \; \hat{\mathbf{n}}.$

• Graphical intuition and a simple example

  • Example: Fishing trip along a straight path A→B with magnitude 6 km northeast.
  • A displacement vector is drawn from A to B with magnitude 6 km and direction northeast; its scale in a drawing depends on chosen units.
  • Opposite displacements along the same path (A→B vs B→A) have the same magnitude but opposite directions (antiparallel): ( \vec{AB} = -\vec{BA} ).

• Parallel, antiparallel, orthogonal relations

  • Antiparallel: same line, opposite directions; written as ( \vec{B} = -\vec{A} ) when directions are opposite.
  • Parallel: same direction (or exactly opposite), vectors may have different magnitudes but lie along the same line.
  • Orthogonal (perpendicular): vectors perpendicular to each other.

• Check Your Understanding 2.1 (conceptual summaries)

  • Compare velocity vectors for various directions and magnitudes to determine equality.

• Vector addition and subtraction (one dimension)

  • The sum of vectors along a single line is simply the algebraic sum of their magnitudes with proper signs.
  • The resultant of two vectors along the same line is also along that line: ( R = A + B ) (parallel vectors).
  • When vectors lie in a plane (2D), use the parallelogram rule or tail-to-head method to construct the resultant.

• Unit vectors and component form (introduction)

  • In 2D, introduce unit vectors along the x- and y-axes: (\hat{\mathbf{i}}) and (\hat{\mathbf{j}}).
  • Any vector in the plane can be written as
    $\vec{v} = v<em>x \hat{\mathbf{i}} + v</em>y \hat{\mathbf{j}},$
    where (vx) and (vy) are scalar components (the projections) of the vector along the x- and y-axes.
  • The magnitude equals
    $|\vec{v}| = \sqrt{v<em>x^2 + v</em>y^2}.$

• Example: a 6 km displacement in a certain direction

  • If a displacement is 6 km at angle (\theta) from the +x-axis, its components are
    $v<em>x = |\vec{v}| \cos\theta, \quad v</em>y = |\vec{v}| \sin\theta,$
    and the angle (\theta) must be chosen according to the quadrant of the vector.

• Displacement and magnitude-angle form

  • The direction angle (\theta) is the angle the vector makes with the +x-axis, measured counterclockwise. In polar form, a vector is represented by ( (|\vec{v}|, \theta) ).

• Example 2.1: Ladybug on a ruler (one-dimensional stick) illustrates cumulative displacements along a single axis with multiple legs of motion; the total displacement is the vector sum of the individual leg vectors, computed via components.

• Summary of 2.1 concepts

  • Scalars vs vectors; vector notation; magnitude and direction; one- and two-dimensional vector algebra; vector components; direction angle; unit vectors; graphical vs analytical methods.

2.2 Coordinate Systems and Components of a Vector

• Two main coordinate systems

  • Cartesian (rectangular) coordinates: use orthogonal axes, typically (x) and (y) in a plane; a point is described by coordinates ((x, y)).
  • Polar coordinates in a plane: radial distance (r) from the origin and angle (\theta) from a chosen reference direction (usually the +x-axis). Angles are measured in radians.

• Vector components in Cartesian coordinates

  • A vector in the plane can be written as
    $\vec{v} = v<em>x \hat{\mathbf{i}} + v</em>y \hat{\mathbf{j}}.$
  • The x- and y-components are the projections of the vector onto the axes. They can be found either as the projection of (\vec{v}) or as the difference of coordinates if the vector is defined by its start and end points:
  • If the vector begins at (b=(xb,yb)) and ends at (e=(xe,ye)), then
    $v<em>x = x</em>e - x<em>b, \quad v</em>y = y<em>e - y</em>b.$
  • The magnitude in 2D is
    $|\vec{v}| = \sqrt{v<em>x^2 + v</em>y^2}.$
  • The vector is the diagonal of a rectangle whose sides are the components along the axes.

• Example 2.3: Displacement of a mouse pointer

  • Given initial position (b=(6.0\,\text{cm}, 1.6\,\text{cm})) and end position (e=(2.0\,\text{cm}, 4.5\,\text{cm})).
  • Displacement components:
    $\Delta x = x<em>e - x</em>b = 2.0 - 6.0 = -4.0\ \text{cm},$
    $\Delta y = y<em>e - y</em>b = 4.5 - 1.6 = 2.9\ \text{cm}.$
  • Magnitude: $|\vec{v}| = \sqrt{(-4.0)^2 + (2.9)^2} \approx 4.94\ \text{cm}.$
  • Direction: angle measured from +x-axis, using (\tan\theta = \Delta y / \Delta x) with quadrant adjustment; here (\theta \approx 144^\circ).

• Polar coordinates and Cartesian conversion

  • For a point P with polar coordinates ((r, \theta)):
    $x = r \cos\theta, \quad y = r \sin\theta$
  • Conversely, ( r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right) ) with quadrant adjustments.

• Vectors in three dimensions

  • A 3D vector is written as
    $\vec{v} = v<em>x \hat{\mathbf{i}} + v</em>y \hat{\mathbf{j}} + v_z \hat{\mathbf{k}}.$
  • The magnitude generalizes to
    $|\vec{v}| = \sqrt{v<em>x^2 + v</em>y^2 + v_z^2}.$
  • A right-handed Cartesian system uses the standard orientation: x, y, z axes in a right-handed sequence.

• Polar and spherical viewpoints

  • In 3D, the orientation becomes more complex; often one uses components along x/y/z or uses a spherical representation with radial distance, polar angle, and azimuthal angle.

• Examples in 2D and 3D

  • Example 2.6: Polar coordinates for locating objects; relation to rectangular coordinates; converting between coordinate systems.
  • Example 2.7: Takeoff of a drone (3D coordinates): position at two times, displacement, and magnitude.

• Unit vectors and direction

  • A unit vector provides a pure direction without magnitude, essential when expressing a direction vector as a linear combination with a magnitude.
  • The unit vector of direction for any vector is
    $\hat{\mathbf{u}} = \frac{\vec{v}}{|\vec{v}|}.$

• Significance of coordinate systems

  • Cartesian coordinates are convenient for displacements and forces; polar coordinates simplify rotational descriptions; 3D Cartesian is standard for space problems and orientation.

• Example connections and practice problems

  • Example 2.7 (Drone takeoff) demonstrates computing a displacement vector from two 3D positions.
  • Check Your Understanding 2.4, 2.5, 2.6 emphasize applying components and polar/rectangular conversions.

2.3 Algebra of Vectors

• Core ideas

  • Vectors can be added to other vectors and multiplied by scalars. Vector addition is associative and commutative; scalar multiplication is distributive over both vector addition and scalar addition.
  • Vector equations have the same form as scalar equations, but each term is a vector.
  • The null (zero) vector (\vec{0}) has all components zero and acts as the additive identity.
  • Two vectors are equal if and only if all corresponding components are equal: (\vec{A} = \vec{B} \iff Ax=Bx, Ay=By, Az=Bz).

• Resultant of a sum of vectors

  • If (\vec{R} = \vec{A} + \vec{B} + \dots), then component-wise:
    $R<em>x = A</em>x + B<em>x + \dots, \quad R</em>y = A<em>y + B</em>y + \dots, \quad R<em>z = A</em>z + B_z + \dots.$
  • The magnitude is then $|\vec{R}| = \sqrt{R<em>x^2 + R</em>y^2 + R_z^2}.$

• Analytical vs graphical methods

  • Analytical methods (component-wise) give exact results; graphical methods (parallelogram, tail-to-head) are useful for visualization and quick approximations.
  • The distributive law can be used to handle sums of many vectors efficiently.

• Example 2.9: Analytical computation of a resultant

  • Given three vectors in a plane with magnitudes and directions, resolve into scalar components, then sum components to obtain the resultant vector.
  • After obtaining the scalar components, express the resultant in vector form and compute magnitude/direction if needed.

• Examples with multiple vectors

  • Vacation trek example with multiple displacements demonstrates repeated parallelogram applications or a direct tail-to-head construction to yield the same resultant, illustrating associativity and commutativity.

• Applications in physics

  • Kinematics: resultant displacement/velocity; Mechanics: resultant force; Electricity & Magnetism: resultant fields.

• Example 2.11 and 2.12 demonstrate solving vector equations by resolving into components, grouping by axes, and then reconstructing the vector.

• Unit vectors and components in practice

  • When vectors are given in unit-vector form, cross-check using distributive and commutative properties:
    $\vec{A} = A<em>x \hat{\mathbf{i}} + A</em>y \hat{\mathbf{j}} + A_z \hat{\mathbf{k}}.$
  • The choice of order in summing vectors does not affect the final resultant due to associativity/commutativity of vector addition.

2.4 Products of Vectors

• Two kinds of products

  • Scalar product (dot product): yields a scalar.
  • Vector product (cross product): yields a vector perpendicular to both operands.
  • Note: Vectors can be multiplied by scalars, but vectors cannot be divided by vectors; division is defined only by multiplying by a reciprocal scalar.

• The scalar (dot) product

  • Definition: For vectors (\vec{A}) and (\vec{B}) with angle (\theta) between them,
    $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos\theta.$
  • In components: if (\vec{A} = Ax \hat{\mathbf{i}} + Ay \hat{\mathbf{j}} + Az \hat{\mathbf{k}}) and similarly for (\vec{B}), then $\vec{A} \cdot \vec{B} = A</em>x B<em>x + A</em>y B<em>y + A</em>z B_z.$
  • Key properties:
  • Commutative: (\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}).
  • Distributive: (\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}).
  • Orthogonal vectors: if (\vec{A} \perp \vec{B}) then (\vec{A} \cdot \vec{B} = 0).
  • Self-dot product: $\vec{A} \cdot \vec{A} = |\vec{A}|^2.$
  • Geometric interpretation: projection of one vector onto the other times the magnitude of the second vector; equivalently the magnitude of the component of (\vec{A}) in the direction of (\vec{B}) times (|\vec{B}|).
  • Unit vectors: in Cartesian coordinates, dot products with axis unit vectors yield scalar components (e.g., the x-component is the dot product with (\hat{\mathbf{i}})).

• The vector (cross) product

  • Definition: The cross product of two vectors is a vector
    $\vec{A} \times \vec{B}$
    that is perpendicular to the plane containing (\vec{A}) and (\vec{B}).
  • Magnitude:$|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin\theta,$ where (\theta) is the angle from (\vec{A}) to (\vec{B}).
  • Direction: given by the corkscrew (right-hand) rule: align your right hand with the rotation from (\vec{A}) to (\vec{B}); the thumb points in the direction of the cross product.
  • Non-commutativity: (\vec{A} \times \vec{B} = - (\vec{B} \times \vec{A}).)
  • Vanishing cross product: if (\vec{A}) and (\vec{B}) are parallel or antiparallel, then (\vec{A} \times \vec{B} = \vec{0}.)
  • In Cartesian coordinates, the cross product can be computed via the determinant form:
    \vec{A} \times \vec{B} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \[2pt] Ax & Ay & Az \[2pt] Bx & By & Bz \end{vmatrix} = (Ay Bz - Az By)\hat{\mathbf{i}} + (Az Bx - Ax Bz)\hat{\mathbf{j}} + (Ax By - Ay Bx)\hat{\mathbf{k}}.

• Unit vectors and cross products

  • The fundamental cross products among unit vectors are:
    $\hat{\mathbf{i}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}}, \quad \hat{\mathbf{j}} \times \hat{\mathbf{k}} = \hat{\mathbf{i}}, \quad \hat{\mathbf{k}} \times \hat{\mathbf{i}} = \hat{\mathbf{j}}.$
  • Reversing the order changes the sign: e.g., (\hat{\mathbf{j}} \times \hat{\mathbf{i}} = -\hat{\mathbf{k}}).
  • Similar rules apply for cross products involving generic vectors via distributivity.

• Applications of dot and cross products

  • Work done by a force: if a force (\vec{F}) moves an object by displacement (\vec{d}), the work is
    $W = \vec{F} \cdot \vec{d}.$
  • Torque: torque is the cross product of the lever arm (position vector) and the force: $\boldsymbol{\tau} = \vec{r} \times \vec{F}.$
  • Magnetic force on a moving charge: $\vec{F} = q \; \vec{v} \times \vec{B}.$ For example, the magnitude and direction can be found by either component methods or by dot/cross product formulas; the force is always perpendicular to the magnetic field in the simple case where (|\vec{v}|) and (|\vec{B}|) are given.

• Example 2.18: Torque of a force on a wrench

  • Visualizes torque as a cross product between the lever arm and the applied force; direction given by corkscrew rule; magnitude is
    $|\boldsymbol{\tau}| = |\vec{r} \times \vec{F}| = |\vec{r}| |\vec{F}| \sin\phi.$
  • The maximum torque occurs when the force is perpendicular to the wrench handle ((\phi=90^\circ)).

• Summary of 2.4 concepts

  • Dot product yields a scalar; cross product yields a vector.
  • Dot product useful for projections, work, and energy relations; cross product useful for rotations, torques, and magnetic forces.
  • The two products obey different algebraic rules (commutative vs anticommutative, etc.).

Quick Reference: Key Equations (from the sections above)

• Vector as component form (2D Cartesian):
  $\vec{v} = v<em>x \hat{\mathbf{i}} + v</em>y \hat{\mathbf{j}}.$
  Magnitude: $|\vec{v}| = \sqrt{v<em>x^2 + v</em>y^2}.$
  Angle from +x: $\theta = \tan^{-1}\left(\frac{v<em>y}{v</em>x}\right)\,,$ with quadrant awareness.

• 3D vector form:
  $\vec{v} = v<em>x \hat{\mathbf{i}} + v</em>y \hat{\mathbf{j}} + v<em>z \hat{\mathbf{k}}.$ Magnitude: $|\vec{v}| = \sqrt{v</em>x^2 + v<em>y^2 + v</em>z^2}.$

• Dot product (scalar product):
  $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos\theta = A<em>x B</em>x + A<em>y B</em>y + A<em>z B</em>z.$
  Properties: commutative, distributive; orthogonality yields zero; (\vec{A} \cdot \vec{A} = |\vec{A}|^2).

• Cross product (vector product):
  $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{\mathbf{i}} &amp; \hat{\mathbf{j}} &amp; \hat{\mathbf{k}} \ A<em>x & A</em>y &amp; A<em>z \ B</em>x &amp; B<em>y & B</em>z \end{vmatrix} = (A<em>y B</em>z - A<em>z B</em>y)\hat{\mathbf{i}} + (A<em>z B</em>x - A<em>x B</em>z)\hat{\mathbf{j}} + (A<em>x B</em>y - A<em>y B</em>x)\hat{\mathbf{k}}.$
  Magnitude: $|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin\theta.$
  Direction: given by corkscrew right-hand rule; anticommutative: (\vec{A} \times \vec{B} = - (\vec{B} \times \vec{A})).

• Displacement vector from coordinates
  $\vec{d} = (x<em>e - x</em>b) \hat{\mathbf{i}} + (y<em>e - y</em>b) \hat{\mathbf{j}} + (z<em>e - z</em>b) \hat{\mathbf{k}}.$

• Unit vector of direction
  $\hat{\mathbf{u}} = \frac{\vec{v}}{|\vec{v}|}.$

• Magnitude of a vector from components
  $|\vec{v}| = \sqrt{v<em>x^2 + v</em>y^2 + v_z^2}.$

• Polar to Cartesian conversion (plane)
  $x = r \cos\theta, \quad y = r \sin\theta.$

• Cartesian to polar conversion (plane)
  $r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)$ (with quadrant adjustment).

• Work done by a force
  $W = \vec{F} \cdot \vec{d}.$

• Torque
  $\boldsymbol{\tau} = \vec{r} \times \vec{F}.$

• Magnetic force on a moving charge
  $\vec{F} = q \, \vec{v} \times \vec{B}.$

Notes:

• The above notes summarize the major and minor points from the transcript, including definitions, geometric interpretations, principal equations, and representative examples. They are organized to serve as a comprehensive study aid that can replace the original source for preparation purposes.