Vectors and Scalars

Vector vs. Scalar Quantities

• Definitions
  • Vector: Numerical quantity possessing both magnitude and direction.
  • Scalar: Numerical quantity possessing magnitude only (no direction).
• Common examples
  • Vectors: displacement, velocity, acceleration, force.
  • Scalars: distance, speed, energy, pressure, mass.
• Why the distinction matters
  • On non-linear paths the two differ drastically.
  • Example: Earth’s orbit
  • Annual path length (distance): $9.4\times10^8\ \text{km}$.
  • Net displacement after one full orbit: $0\ \text{km}$ (returns to the starting point).
  • Highlights that distance ≠ displacement when the path loops back.

Representations & Notation

• Graphical: arrows drawn to scale
  • Arrow direction → vector direction.
  • Arrow length → proportional to magnitude.
• Symbolic conventions (Kaplan MCAT series)
  • Boldface ((\mathbf{a})) → vector.
  • Italic ((a)) → magnitude of a vector or a scalar.
• Alternative textbook conventions
  • Arrow over letter ((\vec{a})).
• Direction references
  • Horizontal axis → $x$ component.
  • Vertical axis → $y$ component.
  • Other axes chosen when more convenient (e.g., along an incline).

Vector Addition (Resultant)

• Resultant: the single vector equivalent to adding (or subtracting) others.
• Tip-to-tail method
  1. Keep magnitude & direction constant.
  2. Place tail of (\mathbf{b}) at tip of (\mathbf{a}).
  3. Draw vector from tail of (\mathbf{a}) to tip of (\mathbf{b}) → $\mathbf{a}+\mathbf{b}$.
• Component (analytical) method (often faster for >2 vectors)
  • Break every vector into $x$ and $y$ pieces (or any orthogonal pair).
  • Sum all $x$ components → $R_x$.
  • Sum all $y$ components → $R_y$.
  • Magnitude: $R=\sqrt{R<em>x^2+R</em>y^2}$.
  • Direction: $\theta=\tan^{-1}!\bigl(R<em>y/R</em>x\bigr)$ (inverse tangent; MCAT rarely requires computing it).

Component Decomposition & Reconstruction

• For any vector $\mathbf{v}$ at angle $\theta$ from the $x$-axis:
  $v<em>x = v\cos\theta$$v</em>y = v\sin\theta$
• Conversely, if $v<em>x$ and $v</em>y$ are known:
  $v = \sqrt{v<em>x^2 + v</em>y^2}$
  $\theta = \tan^{-1}!\bigl(v<em>y/v</em>x\bigr)$
• Example (given)
  • v = 10\,\text{m·s}^{-1},\;\theta = 30^{\circ}
  • v_x = 10\cos30^{\circ}=10\cdot\frac{\sqrt3}{2}=5\sqrt3\,\text{m·s}^{-1}
  • v_y = 10\sin30^{\circ}=10\cdot\frac12 = 5\,\text{m·s}^{-1}
• Example (find magnitude)
  • Given vx = 3\,\text{m·s}^{-1},\;vy = 4\,\text{m·s}^{-1}
  • v = \sqrt{3^2+4^2}=\sqrt{25}=5\,\text{m·s}^{-1}

Vector Subtraction

• Concept: $\mathbf{a}-\mathbf{b} = \mathbf{a}+(-\mathbf{b})$, where $-\mathbf{b}$ has the same magnitude as $\mathbf{b}$ but opposite direction.
• Tip-to-tail: flip (\mathbf{b}), then add.
• Component method: subtract components directly
  • $R<em>x = a</em>x - b_x$
  • $R<em>y = a</em>y - b_y$

Multiplying Vectors by Scalars

• New vector $\mathbf{b} = n\mathbf{a}$
  • Magnitude: $|\mathbf{b}| = |n|\,|\mathbf{a}|$.
  • Direction:
  • n>0 → same direction as $\mathbf{a}$.
  • n<0 → opposite direction (antiparallel).
• Examples
  • $3\mathbf{a}$: three times longer, same way.
  • $-3\mathbf{a}$: three times longer, opposite way.

Multiplying Vectors by Vectors

Dot Product (Scalar Product)

• Generates scalar (e.g., work $W$).
• Formula: $\mathbf{a}\cdot\mathbf{b}=ab\cos\theta$.
• Geometric meaning: projection of one vector onto another times the magnitude of the other.
• Significance: captures alignment (max when parallel, zero when perpendicular).

Cross Product (Vector Product)

• Generates vector (e.g., torque $\boldsymbol\tau$).
• Magnitude: $|\mathbf{a}\times\mathbf{b}| = ab\sin\theta$.
• Direction: perpendicular to the plane of $\mathbf{a}$ & $\mathbf{b}$, determined by the Right-Hand Rule (RHR).
• On the 2-D MCAT, this means “into the page” or “out of the page.”

Right-Hand Rule (one common version)

1. Point thumb in direction of $\mathbf{a}$.
2. Extend fingers toward $\mathbf{b}$ (rotate wrist if necessary).
3. Direction your palm faces = direction of $\mathbf{a}\times\mathbf{b}$.

• Alternative mnemonic (index = $\mathbf{a}$, middle = $\mathbf{b}$, thumb = resultant) is equally valid.

Worked Example – Cross Products

• Vectors:
  • $\mathbf{a} = \langle -3\,\text{N},\ 0 \rangle$ (points left along $x$).
  • $\mathbf{b} = \langle 0,\ +4\,\text{m} \rangle$ (points up along $y$).
• Angle between them: $90^{\circ}$.
• Magnitude of $\mathbf{a}\times\mathbf{b}$ and $\mathbf{b}\times\mathbf{a}$:
  ab\sin90^{\circ} = (3)(4)(1) = 12\,\text{N·m}.
• Directions via RHR
  • $\mathbf{c} = \mathbf{a}\times\mathbf{b}$ → “into the page.”
  • $\mathbf{d} = \mathbf{b}\times\mathbf{a}$ → “out of the page.”

Practical & Conceptual Connections

• Physics problem solving: Breaking vectors into components simplifies kinematics, dynamics, E-fields, etc.
• Engineering: Cross product governs rotational quantities (torque, angular momentum).
• Mathematics: Dot & cross products form the backbone of vector calculus and linear algebra; they extend naturally to higher dimensions (though MCAT focuses on 2-D).
• Philosophical/ethical note: Precise notation prevents miscommunication—critical in research, medicine, and engineering safety.