Scalars & Vectors - Comprehensive Study Notes

Scalars

• A scalar is a physical quantity that has only magnitude (no direction).
  • Examples: mass, length, time, temperature, volume, density.
  • Magnitude-only descriptions: physical quantity magnitude; no directional component.
• Numeric examples from transcript:
  • Length of a car is $4.5\ \mathrm{m}$.
  • Net weight: $1000\ \mathrm{g}$.
  • Mass of a gold bar: $1\ \mathrm{kg}$.
  • Time is $12.76\ \mathrm{s}$.
  • Temperature is $36.8^{\circ}\mathrm{C}$.
• Conceptual role: Scalars quantify how much but not how/where in space they act.

Vectors

• A vector is a physical quantity that has both a magnitude and a direction.
  • Examples: position, displacement, velocity, acceleration, momentum, force.
• Representation of a vector:
  • Geometrically: a directed line with a head (tip) and tail.
  • Symbolically: often written as AB (arrow over the letter in print) or as a bold letter (e.g., \mathbf{A}).
  • In text form: a vector is represented by a capital letter with an arrow above it (A⃗ , B⃗ , etc.).
• Common notations:
  • A vector can be denoted as AB or as A⃗; common letters include v (velocity), F (force), r (position), p (momentum).
• Physical significance:
  • Vector quantities convey both how much (magnitude) and in which direction (orientation) they act.

Representation of a vector

• A vector has:
  • Magnitude: length of the vector.
  • Direction: the orientation of the vector in space.
• A vector is often represented by a letter with an arrow: \vec{A}.
• In components: A = Ax î + Ay ĵ + Az k̂ (for 3D) or A = Ax î + A_y ĵ (for 2D).
• A vector can also be written as magnitude times a unit vector: A = |A| \hat{A}.
• Unit vectors (direction only, magnitude = 1):
  • Cartesian unit vectors: î, ĵ, k̂.
  • They satisfy î^2 = ĵ^2 = k̂^2 = 1 and î × ĵ = k̂, etc. (see cross-product rules).
• Negative vectors: reversing a vector reverses its direction; same magnitude.

Unit vectors

• A unit vector is a vector with magnitude exactly 1 and points in the direction of the given vector.
• A vector can be decomposed as: A = |A| \hat{A} where \hat{A} is the unit vector in the direction of A.
• Cartesian unit vectors in 3D:
  • î (along x), ĵ (along y), k̂ (along z).
  • Any vector A can be written as A = Ax î + Ay ĵ + A_z k̂.
• Relationships:
  • If A has components (Ax, Ay, A_z), then the direction cosines are α, β, γ where
  • cos α = A_x / |A|,
  • cos β = A_y / |A|,
  • cos γ = A_z / |A|.
• 2D and 3D components form the basis for resolving vectors along coordinate axes.

Resolution of a vector

• Resolution is splitting a vector into two or more components whose combined effect equals the original vector.
• 2D resolution (into x and y components): A = Ax î + Ay ĵ.
• 2D component relations:
  • Ax = |A| cos θ, Ay = |A| sin θ, where θ is the angle with the x-axis.
• 3D resolution (into x, y, z components): A = Ax î + Ay ĵ + A_z k̂.
• 3D component relations with direction cosines:
  • Ax = |A| cos α, Ay = |A| cos β, A_z = |A| cos γ.
• Magnitude from components:
  • |A| = \sqrt{Ax^2 + Ay^2 + A_z^2}.
• Direction angles from components:
  • α = \cos^{-1}(Ax/|A|), \beta = \cos^{-1}(Ay/|A|), \gamma = \cos^{-1}(A_z/|A|).

Magnitude and direction from components

• For 2D: A = Ax î + Ay ĵ; magnitude |A| = \sqrt{Ax^2 + Ay^2} and direction θ = \tan^{-1}(Ay / Ax).
• For 3D: A = Ax î + Ay ĵ + Az k̂; magnitude |A| = \sqrt{Ax^2 + Ay^2 + Az^2}; direction cosines relate to α, β, γ as above.

Adding vectors by components

• Given A = Ax î + Ay ĵ + Az k̂ and B = Bx î + By ĵ + Bz k̂,
  • R = A + B = (Ax + Bx) î + (Ay + By) ĵ + (Az + Bz) k̂.
• Resultant components: Rx = Ax + Bx, Ry = Ay + By, Rz = Az + B_z.

Multiplying vectors

• There are two primary products:
  • Scalar (dot) product: A · B → scalar.
  • Vector (cross) product: A × B → vector.

Multiplying a vector by a scalar

• If s is a scalar and A is a vector, then sA is a vector with:
  • Magnitude: |sA| = |s| |A|.
  • Direction: same as A if s > 0; opposite if s < 0.
• Examples:
  • 2A, -3A illustrate scaling and possible direction reversal.

Scalar (dot) product

• Definition: A · B = |A||B| cos θ, where θ is the angle between A and B.
• Geometric meaning: product equals the magnitude of A times the component of B along the direction of A.
• In components:
  • If A = Ax î + Ay ĵ + Az k̂ and B = Bx î + By ĵ + Bz k̂, then
  • A · B = Ax Bx + Ay By + Az Bz.
• Examples from transcript:
  • W = F · s (work done) with W = |F||s| cos θ.
  • P = F · v (power or projection concepts): P = F · v = |F||v| cos θ.
• Properties:
  • The scalar product is commutative: A · B = B · A.
  • Distributive over addition: A · (B + C) = A · B + A · C.
  • If A ⟂ B, A · B = 0.
  • If A ∥ B (same direction), A · B = |A||B|.
  • If A ∦ B but anti-parallel, A · B = -|A||B|.
  • The scalar product of a vector with itself: A · A = |A|^2.
  • For unit vectors î, ĵ, k̂: î · î = ĵ · ĵ = k̂ · k̂ = 1; î · ĵ = ĵ · k̂ = k̂ · î = 0.
• Calculating with components:
  • A · B = Ax Bx + Ay By + Az Bz.

Vector (cross) product

• Definition: A × B = |A||B| sin θ n̂, where n̂ is a unit vector perpendicular to the plane containing A and B, in the direction given by the right-hand rule.
• Geometric meaning: magnitude |A × B| equals the area of the parallelogram spanned by A and B; direction is perpendicular to that plane.
• Right-hand rule: align right hand with A and rotate toward B; the thumb points along A × B.
• Applications (examples): τ = r × F (torque); L = r × p (angular momentum).
• Magnitude behavior: |A × B| is maximum when A ⟂ B and zero when A ∥ B or A ∥ −B.
• In Cartesian components (3D):
  • A × B = (Ay Bz - Az By) î + (Az Bx - Ax Bz) ĵ + (Ax By - Ay Bx) k̂.
• Properties:
  • Anti-commutative: A × B = −(B × A).
  • Distributive over addition: A × (B + C) = (A × B) + (A × C).
  • If A or B is a zero vector, A × B = 0.
  • If A ∥ B, A × B = 0.
  • For unit vectors: î × î = 0, î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ; and the cyclic order gives positive results while reversed order gives negative.
• Calculating cross product using components:
  • Let A = Ax î + Ay ĵ + Az k̂ and B = Bx î + By ĵ + Bz k̂, then
  • A × B = (Ay Bz - Az By) î + (Az Bx - Ax Bz) ĵ + (Ax By - Ay Bx) k̂.

Aid to memory and conventions

• Vector notation and unit vectors are crucial to build intuition for projections, perpendicular components, and geometric interpretations.
• The cross product yields a vector perpendicular to the plane of A and B, while the dot product yields a scalar equal to the projection magnitude of one vector onto the other scaled by the magnitude of the other vector.
• Be mindful of sign conventions (positive direction follows right-hand rule; negative values appear for opposite directions).

Summary of core formulas (compact reference)

• Dot product:
  • Definition: $A \cdot B = |A||B| \cos \theta$
  • In components: $A \cdot B = A<em>x B</em>x + A<em>y B</em>y + A<em>z B</em>z$
  • Key cases: perpendicular (0), parallel (|A||B|), anti-parallel (−|A||B|), self-dot: $A\cdot A = |A|^2$
• Cross product:
  • Magnitude: $|A \times B| = |A||B| \sin\theta$
  • Direction: perpendicular to the plane of A and B (right-hand rule)
  • In components: $A \times B = (A<em>y B</em>z - A<em>z B</em>y) î + (A<em>z B</em>x - A<em>x B</em>z) ĵ + (A<em>x B</em>y - A<em>y B</em>x) k̂$
  • Anti-commutative: $A \times B = - (B \times A)$
• Vector addition via components:
  • R = A + B = (Ax + Bx) î + (Ay + By) ĵ + (Az + Bz) k̂
• Vector resolution:
  • 2D: A = Ax î + Ay ĵ with $A<em>x = |A| \cos \theta, \quad A</em>y = |A| \sin \theta$
  • 3D: A = Ax î + Ay ĵ + Az k̂ with $A</em>x = |A| \cos \alpha, \quad A<em>y = |A| \cos \beta, \quad A</em>z = |A| \cos \gamma$
• Direction angles from components:
  • $\alpha = \cos^{-1}\left(\frac{A<em>x}{|A|}\right), \quad \beta = \cos^{-1}\left(\frac{A</em>y}{|A|}\right), \quad \gamma = \cos^{-1}\left(\frac{A_z}{|A|}\right)$
• Case examples for vector addition (magnitude and direction):
  • Case I (θ = 0°, parallel same direction): magnitude $|R| = P + Q$; direction same as the larger of P and Q (assuming positive magnitudes).
  • Case II (θ = 90°, perpendicular): magnitude $|R| = \sqrt{P^2 + Q^2}$; direction angle $\alpha = \tan^{-1}\left(\frac{Q}{P}\right)$ (relative to P).
  • Case III (θ = 180°, anti-parallel): magnitude $|R| = |P - Q|$; direction depends on which of P or Q is larger, effectively aligned with the larger magnitude.
• Vector representation in 3D using unit vectors:
  • A = |A| \hat{A} = Ax î + Ay ĵ + A_z k̂,
  • where Ax, Ay, A_z are the projections on the axes and relate to direction cosines as above.