Comprehensive Vector Algebra for Medical Physics

Fundamental Definitions of Scalars and Vectors

  • Vector Algebra Overview: Vector algebra serves as a critical mathematical foundation in physics. Physical quantities possessing both a size (magnitude) and a direction are represented as vectors. Proficiency in this area is required for mechanics, electromagnetism, fluid mechanics, quantum physics, and other branches of study.

  • Definition of a Vector: A vector is a physical quantity characterized by two necessary components:

    • Magnitude: The size or numerical value.

    • Direction: The orientation of the quantity in space.

    • Examples: Force, Velocity, Acceleration, Displacement, Momentum, Electric field, and Magnetic field.

    • Example Illustration: A force specified as 20 N20\,N east is a vector because it contains both a magnitude (20 N20\,N) and a direction (East). Omitting the direction leaves the force incompletely defined.

  • Definition of a Scalar: A scalar is a physical quantity defined exclusively by its magnitude, with no direction involved.

    • Examples: Mass, Charge, Density, Temperature, Time, Energy, Distance, and Speed.

    • Example Illustration: A mass of 10 kg10\,kg is fully specified because direction is irrelevant to the quantity.

  • Comparative Analysis:

    • Scalars: Possess magnitude only; added via ordinary arithmetic; no direction involved. Examples: mass, energy, time.

    • Vectors: Possess both magnitude and direction; added using specific vector laws; direction is essential. Examples: force, velocity, displacement.

Basic Vector Operations

  • Vector Addition: The resultant (sum) of two vectors a⃗\vec{a} and b⃗\vec{b} is denoted as R⃗=a⃗+b⃗\vec{R} = \vec{a} + \vec{b}. Unlike scalar addition, magnitudes cannot simply be summed; direction must be accounted for.

    • Head-to-tail Method:

      1. Draw the first vector.

      2. Position the tail of the second vector at the head of the first vector.

      3. The resultant is the vector drawn from the tail of the first vector to the head of the second vector, representing the total combined effect.

  • Vector Subtraction: Subtracting a vector is mathematically defined as adding its negative: a⃗−b⃗=a⃗+(−b⃗)\vec{a} - \vec{b} = \vec{a} + (-\vec{b}).

    • The Negative Vector: A negative vector (−b⃗-\vec{b}) has the exact same magnitude as b⃗\vec{b} but points in the opposite direction.

  • Properties of Vector Addition:

    1. Commutative Law: The order of addition is irrelevant (a⃗+b⃗=b⃗+a⃗\vec{a} + \vec{b} = \vec{b} + \vec{a}).

    2. Associative Law: The way vectors are grouped does not change the sum (a⃗+(b⃗+c⃗)=(a⃗+b⃗)+c⃗\vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c}).

  • General Notational Rules:

    • Vectors are symbolized by bold letters or symbols with an arrow above them (e.g., a⃗\vec{a}).

    • The length of a vector in a diagram represents its magnitude.

    • The arrowhead denotes the direction.

    • Two vectors are considered equal if and only if they share the same magnitude and direction, independent of their spatial location.

Vector Components and Cartesian Coordinates

  • The Cartesian Coordinate System: This system uses three mutually perpendicular axes: the xx-axis, yy-axis, and zz-axis. Resolving vectors into components along these axes simplifies operations like addition, subtraction, and calculus.

  • Mathematical Representation: A three-dimensional vector is the sum of its components:

    • a⃗=a⃗x+a⃗y+a⃗z\vec{a} = \vec{a}_x + \vec{a}_y + \vec{a}_z

    • a⃗x\vec{a}_x: Component along the xx-axis.

    • a⃗y\vec{a}_y: Component along the yy-axis.

    • a⃗z\vec{a}_z: Component along the zz-axis.

  • Magnitude (Length) of a Vector: Denoted as ∣a⃗∣|\vec{a}| or simply aa. Derived from the three-dimensional Pythagorean theorem:

    • a=ax2+ay2+az2a = \sqrt{a_x^2 + a_y^2 + a_z^2}

    • Example 1: For a vector with components ax=3a_x = 3, ay=4a_y = 4, and az=12a_z = 12:         a=32+42+122=9+16+144=169=13a = \sqrt{3^2 + 4^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13

  • Two-Dimensional Vectors: Frequently used for planar motion.

    • Expression: a⃗=a⃗x+a⃗y\vec{a} = \vec{a}_x + \vec{a}_y

    • Magnitude: a=ax2+ay2a = \sqrt{a_x^2 + a_y^2}

    • Trigonometric Resolution: If a vector makes an angle ϕ\phi with the positive xx-axis:

      • Horizontal component: ax=acos⁡(ϕ)a_x = a \cos(\phi)

      • Vertical component: ay=asin⁡(ϕ)a_y = a \sin(\phi)

    • Direction Determination: If components are known, the angle is found via:         tan⁡(ϕ)=ayax  ⟹  ϕ=tan⁡−1(ayax)\tan(\phi) = \frac{a_y}{a_x} \implies \phi = \tan^{-1}\left(\frac{a_y}{a_x}\right)         Note: The specific quadrant must be considered as the inverse tangent may not uniquely identify the angle.

    • Example 2: A vector with magnitude a=20a = 20 at 30∘30^\circ to the positive xx-axis:         ax=20cos⁡(30∘)=20(32)=17.32a_x = 20 \cos(30^\circ) = 20\left(\frac{\sqrt{3}}{2}\right) = 17.32         ay=20sin⁡(30∘)=20(0.5)=10a_y = 20 \sin(30^\circ) = 20(0.5) = 10         Resulting vector: a⃗=17.32i^+10j^\vec{a} = 17.32\hat{i} + 10\hat{j}

Unit Vectors

  • Purpose: A unit vector specifies direction only and has a magnitude of exactly 1. For any vector a⃗\vec{a}, the unit vector a^\hat{a} is:

    • a^=a⃗∣a⃗∣\hat{a} = \frac{\vec{a}}{|\vec{a}|}

  • Standard Cartesian Unit Vectors:

    • i^\hat{i}: Positive xx-axis direction.

    • j^\hat{j}: Positive yy-axis direction.

    • k^\hat{k}: Positive zz-axis direction.

    • Magnitudes: ∣i^∣=∣j^∣=∣k^∣=1|\hat{i}| = |\hat{j}| = |\hat{k}| = 1

  • Algebraic Vector Form: Standard notation in university-level physics and engineering uses components and unit vectors:

    • a⃗=axi^+ayj^+azk^\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}

Alternative Coordinate Systems

  • Selection Criteria: In electromagnetism and other fields, coordinate systems are chosen based on the symmetry of the problem:

    • Cylindrical Symmetry: Used for long straight wires.

    • Spherical Symmetry: Used for point charges.

    • Polar Coordinates: Used for motion in a plane.

  • Polar Coordinates (2D):

    • Variables: Radial distance rr (distance from origin) and angular position θ\theta (angle from positive xx-axis).

    • Vector Form: a⃗=arr^+aθθ^\vec{a} = a_r\hat{r} + a_{\theta}\hat{\theta}

    • Unit Vectors: r^\hat{r} points away from the origin; θ^\hat{\theta} is perpendicular to r^\hat{r}, pointing in the direction of increasing θ\theta. These are "position-dependent" unit vectors as they change direction based on coordinates.

    • Conversions and Relationships:

      • x=rcos⁡(θ)x = r \cos(\theta)

      • y=rsin⁡(θ)y = r \sin(\theta)

      • r=x2+y2r = \sqrt{x^2 + y^2}

      • θ=tan⁡−1(yx)\theta = \tan^{-1}\left(\frac{y}{x}\right)

    • Applications: Motion of charges in circular paths, rotating electric fields, and planar wave propagation.

  • Cylindrical Coordinates (3D):

    • Variables: (r,θ,z)(r, \theta, z). Vector form: a⃗=arr^+aθθ^+azz^\vec{a} = a_r\hat{r} + a_{\theta}\hat{\theta} + a_z\hat{z}.

    • Conversions: x=rcos⁡(θ)x = r \cos(\theta), y=rsin⁡(θ)y = r \sin(\theta), z=zz = z.

    • Advantages: Mathematics is simplified for systems where fields depend only on distance from a center line, such as long-charged wires (E=f(r)E = f(r)).

    • Applications: Coaxial cables, solenoids, cylindrical capacitors, magnetic fields of straight conductors, and plasma physics.

  • Spherical Coordinates (3D):

    • Variables: (r,θ,ϕ)(r, \theta, \phi) where rr is distance from origin, θ\theta is the polar angle, and ϕ\phi is the azimuthal angle.

    • Vector form: a⃗=arr^+aθθ^+aϕϕ^\vec{a} = a_r\hat{r} + a_{\theta}\hat{\theta} + a_{\phi}\hat{\phi}.

    • Conversions:

      • x=rsin⁡(θ)cos⁡(ϕ)x = r \sin(\theta) \cos(\phi)

      • y=rsin⁡(θ)sin⁡(ϕ)y = r \sin(\theta) \sin(\phi)

      • z=rcos⁡(θ)z = r \cos(\theta)

    • Advantages: Ideal for Coulomb's Law and point charges because fields are often perfectly radial (E⃗=Err^\vec{E} = E_r\hat{r}) with no angular components.

    • Applications: Planets, stars, atomic nuclei, Gauss's Law, gravitational fields, and quantum mechanics.

Scalar (Dot) Product

  • Definition: The dot product of vectors A⃗\vec{A} and B⃗\vec{B} results in a scalar quantity:

    • A⃗⋅B⃗=ABcos⁡(θ)\vec{A} \cdot \vec{B} = AB \cos(\theta)

    • A=∣A⃗∣,B=∣B⃗∣A = |\vec{A}|, B = |\vec{B}|, and θ\theta is the angle between them.

  • Physical Meaning: It measures how much of one vector acts in the direction of the other.

    • Parallel Vectors (θ=0∘\theta = 0^\circ): Product is maximum (A⃗⋅B⃗=AB\vec{A} \cdot \vec{B} = AB).

    • Perpendicular Vectors (θ=90∘\theta = 90^\circ): Product is zero (A⃗⋅B⃗=0\vec{A} \cdot \vec{B} = 0).

    • Opposite Vectors (θ=180∘\theta = 180^\circ): Product is negative maximum (A⃗⋅B⃗=−AB\vec{A} \cdot \vec{B} = -AB).

  • Cartesian Form: A⃗⋅B⃗=AxBx+AyBy+AzBz\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z.

  • Properties:

    1. Commutative: A⃗⋅B⃗=B⃗⋅A⃗\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}

    2. Distributive: A⃗⋅(B⃗+C⃗)=A⃗⋅B⃗+A⃗⋅C⃗\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}

    3. Scalar Multiplication: (kA⃗)⋅B⃗=k(A⃗⋅B⃗)(k\vec{A}) \cdot \vec{B} = k(\vec{A} \cdot \vec{B})

    4. Derivative: ddt(A⃗⋅B⃗)=A⃗⋅dB⃗dt+B⃗⋅dA⃗dt\frac{d}{dt}(\vec{A} \cdot \vec{B}) = \vec{A} \cdot \frac{d\vec{B}}{dt} + \vec{B} \cdot \frac{d\vec{A}}{dt}

  • Unit Vector Dot Products:

    • i^⋅i^=1,j^⋅j^=1,k^⋅k^=1\hat{i} \cdot \hat{i} = 1, \hat{j} \cdot \hat{j} = 1, \hat{k} \cdot \hat{k} = 1

    • i^⋅j^=0,i^⋅k^=0,j^⋅k^=0\hat{i} \cdot \hat{j} = 0, \hat{i} \cdot \hat{k} = 0, \hat{j} \cdot \hat{k} = 0

  • Example: For A⃗=2i^+3j^−4k^\vec{A} = 2\hat{i} + 3\hat{j} - 4\hat{k} and B⃗=5i^−2j^+k^\vec{B} = 5\hat{i} - 2\hat{j} + \hat{k}:     A⃗⋅B⃗=(2)(5)+(3)(−2)+(−4)(1)=10−6−4=0\vec{A} \cdot \vec{B} = (2)(5) + (3)(-2) + (-4)(1) = 10 - 6 - 4 = 0     The vectors are perpendicular.

Vector (Cross) Product

  • Definition: The cross product of two vectors A⃗\vec{A} and B⃗\vec{B} results in a third vector perpendicular to both:

    • A⃗×B⃗=ABsin⁡(θ)n^\vec{A} \times \vec{B} = AB \sin(\theta) \hat{n}

    • n^\hat{n} is the unit vector perpendicular to the plane formed by A⃗\vec{A} and B⃗\vec{B}.

  • Right-Hand Rule for Direction:

    1. Point right-hand fingers along A⃗\vec{A}.

    2. Rotate fingers toward B⃗\vec{B}.

    3. The thumb indicates the direction of A⃗×B⃗\vec{A} \times \vec{B}.

  • Special Cases:

    • Parallel Vectors (θ=0∘\theta = 0^\circ): Cross product is zero.

    • Perpendicular Vectors (θ=90∘\theta = 90^\circ): Magnitude is maximum (∣A⃗×B⃗∣=AB|\vec{A} \times \vec{B}| = AB).

  • Unit Vector Cross Products:

    • i^×j^=k^\hat{i} \times \hat{j} = \hat{k}

    • j^×k^=i^\hat{j} \times \hat{k} = \hat{i}

    • k^×i^=j^\hat{k} \times \hat{i} = \hat{j}

    • Reversing Order: j^×i^=−k^\hat{j} \times \hat{i} = -\hat{k}, k^×j^=−i^\hat{k} \times \hat{j} = -\hat{i}, i^×k^=−j^\hat{i} \times \hat{k} = -\hat{j}.

    • Same Vectors: i^×i^=0,j^×j^=0,k^×k^=0\hat{i} \times \hat{i} = 0, \hat{j} \times \hat{j} = 0, \hat{k} \times \hat{k} = 0.

  • Properties:

    • Anticommutative: A⃗×B⃗=−(B⃗×A⃗)\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})

    • Distributive: A⃗×(B⃗+C⃗)=A⃗×B⃗+A⃗×C⃗\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}

    • Derivative: ddt(A⃗×B⃗)=A⃗×dB⃗dt+dA⃗dt×B⃗\frac{d}{dt}(\vec{A} \times \vec{B}) = \vec{A} \times \frac{d\vec{B}}{dt} + \frac{d\vec{A}}{dt} \times \vec{B}

  • Applications: Torque (moment of force), angular momentum, magnetic force, and rotational motion.

Vector and Scalar Fields

  • Intro to Fields: A field exists when a quantity varies point-to-point in space.

  • Vector Field (Physics Context):

    • Definition: A function F⃗(x,y,z)\vec{F}(x, y, z) that assigns a specific vector to every position in space.

    • Graphical Representation: Drawn using arrows where the length indicates strength (magnitude) and the arrow points in the field's direction.

    • Examples:

      1. Velocity Field: v⃗=v⃗(x,y,z)\vec{v} = \vec{v}(x, y, z) in fluid mechanics.

      2. Electric Field: E⃗=E⃗(x,y,z)\vec{E} = \vec{E}(x, y, z) in electrostatics.

      3. Magnetic Field: B⃗=B⃗(x,y,z)\vec{B} = \vec{B}(x, y, z).

      4. Gravitational Field: g⃗=g⃗(x,y,z)\vec{g} = \vec{g}(x, y, z).

  • Scalar Field:

    • Definition: A function U(x,y,z)U(x, y, z) that assigns a single numerical value (magnitude only) to every point.

    • Examples: Temperature distribution T(x,y,z)T(x, y, z), Pressure distribution P(x,y,z)P(x, y, z), Electric potential V(x,y,z)V(x, y, z), and Density ρ(x,y,z)\rho(x, y, z).

  • Comparative Summary:

    • Scalar Field: Magnitude only; one value per point. Examples: temperature, potential.

    • Vector Field: Magnitude and direction; one vector per point. Examples: magnetic and electric fields.

Advanced Geometric Vectors

  • Tangential Vector (dl⃗d\vec{l}):

    • Definition: An infinitesimal displacement along a smooth curve CC that touches the curve without crossing it.

    • Direction: Points along the tangent of the curve; describes instantaneous motion.

    • Applications: Work calculations (∫F⃗⋅dl⃗\int \vec{F} \cdot d\vec{l}), potential, and Ampere's/Faraday's laws.

  • Surface Vector (da⃗d\vec{a}):

    • Definition: A vector associated with a small area segment dada.

    • Attributes: Magnitude equals the area dada; direction is perpendicular (normal) to the surface (∣da⃗∣=da|d\vec{a}| = da).

    • Unit Normal Expression: da⃗=n^dad\vec{a} = \hat{n} da, where n^\hat{n} is perpendicular to the surface.

    • Flux: Surface vectors are essential for calculating field flow through a surface. Electric flux ΦE=∫E⃗⋅da⃗\Phi_E = \int \vec{E} \cdot d\vec{a}; Magnetic flux ΦB=∫B⃗⋅da⃗\Phi_B = \int \vec{B} \cdot d\vec{a}. The dot product ensures only the normal component of the field is counted toward the flux.

Questions & Discussion

  • Q1: Classification

    • Classify as scalar or vector: (a) Pressure (b) Electric Field (c) Density (d) Velocity (e) Work.

  • Q2: Current vs. Current Density

    • (a) Why is electric current not a true vector?

    • (b) What property allows current density (j⃗\vec{j}) to be a vector?

  • Q3: Resultant Proof

    • Add A⃗\vec{A} (5 units East) and B⃗\vec{B} (5 units North). (a) Use diagrams to prove A⃗+B⃗=B⃗+A⃗\vec{A} + \vec{B} = \vec{B} + \vec{A}. (b) Calculate magnitude and direction of the resultant.

  • Q4: Magnitude Thresholds

    • For $|\vec{P}| = 6andand|\vec{Q}| = 8: (a) Find max magnitude and angle. (b) Find min magnitude and angle.\n\n* **Q5: Unit Vector Representation**\n * Vector \vec{A}hasmagnitude25,has magnitude 25,\text{angle} = 36.87^\circabovepositiveabove positivex−axis.(a)Expressin-axis. (a) Express in\hat{i}andand\hat{j}.(b)Findunitvector. (b) Find unit vector\hat{a}.\n\n* **Q6: Ant Displacement**\n * Ant moves from A(1, -2, 3)totoB(4, 2, -1).(a)Finddisplacementvector. (a) Find displacement vector\vec{r}_{AB}. (b) Find straight-line distance.\n\n* **Q7: Dot Product Calculation**\n * Given \vec{A} = 3\hat{i} - 4\hat{j}andand\vec{B} = 2\hat{i} + 6\hat{j}.(a)Find. (a) Find\vec{A} \cdot \vec{B}.(b)Findangle. (b) Find angle\phi. (c) Determine if mostly parallel or anti-parallel.\n\n* **Q8: Cross Product Verification**\n * Given \vec{A} = 1\hat{i} + 2\hat{j} + 3\hat{k}andand\vec{B} = 4\hat{i} + 5\hat{j} + 6\hat{k}.(a)Calculate. (a) Calculate\vec{C} = \vec{A} \times \vec{B}.(b)Verify. (b) Verify\vec{A} \cdot \vec{C} = 0. (c) State geometric meaning of cross product magnitude.\n\n* **Q9: Field Definitions**\n * (a) Define vector field \vec{F}vsscalarfieldvs scalar fieldu. (b) Give future examples from physics curriculum.\n\n* **Q10: Field Calculations**\n * For \vec{F}(x, y) = (2xy)\hat{i} + (x^2)\hat{j}andandV(x, y) = x^2y:(a)Findfieldvalueat: (a) Find field value atP(2, 3).(b)Findscalarvalueat. (b) Find scalar value atP(2, 3).(c)Determinewhatphysicalquantity. (c) Determine what physical quantity\vec{F}representsforachargeatrepresents for a charge atP.\n\n* **Q11: Curve Integration**\n * (a) State direction of d\vec{l}forcurvefor curveC.(b)Determinetotaldisplacementfortheclosedintegral. (b) Determine total displacement for the closed integral\oint d\vec{l}.\n\n* **Q12: Surface Vector Conventions**\n * (a) Direction of d\vec{a}$$ for closed surfaces (e.g., spheres). (b) Direction for open surfaces (e.g., disks) using right-hand rule with counter-clockwise rotation.