Physics and Mathematics

The cross product $\vec{a} \times \vec{b}$ is:

Magnitude: $|\vec{a} \times \vec{b}| = ab\sin\theta$ where $\theta$ is the smaller angle between vectors

Direction: Perpendicular to the plane containing both vectors, following the right-hand thumb rule (fingers along $\vec{a}$, curl toward $\vec{b}$, thumb gives direction)

In component form: $\vec{a} \times \vec{b} = (a_y b_z - a_z b_y)\hat{i} + (a_z b_x - a_x b_z)\hat{j} + (a_x b_y - a_y b_x)\hat{k}$

Key properties:

• $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$ (non-commutative)
• The cross product of parallel vectors is zero
• In a right-handed coordinate system: $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$

The derivative $\frac{dy}{dx}$ is defined as: $\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$

As a rate measurer:

• Represents the instantaneous rate of change of y with respect to x
• Equals the slope of the tangent to the curve y-x at the point of interest
• For a small change Δx, the corresponding change in y can be approximated as: $\Delta y \approx \frac{dy}{dx}\Delta x$

Interpretation:

• Positive $\frac{dy}{dx}$ means y increases as x increases
• Negative $\frac{dy}{dx}$ means y decreases as x increases
• The magnitude of $\frac{dy}{dx}$ indicates how rapidly y changes relative to x

Example: If position s = f(t), then $\frac{ds}{dt}$ gives the instantaneous velocity at time t.

Vector addition in component form works by:

1. Adding the corresponding components separately:

   • For vectors $\vec{a} = a_x\hat{\imath} + a_y\hat{\jmath}$ and $\vec{b} = b_x\hat{\imath} + b_y\hat{\jmath}$
   • Their sum $\vec{c} = \vec{a} + \vec{b} = (a_x + b_x)\hat{\imath} + (a_y + b_y)\hat{\jmath}$

2. This reveals key properties about vector operations:

   • Vector addition is commutative: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$
   • Vector addition is associative: $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$
   • Component-wise addition makes complex vector operations manageable

Example: A boat moving at 3 m/s east (x-component) and 2 m/s north (y-component) while in a current flowing at 1 m/s east and 2 m/s south would have a resultant velocity of $(3+1)\hat{\imath} + (2-2)\hat{\jmath} = 4\hat{\imath}$ m/s, or 4 m/s directly eastward.

A local maximum of a function f(x) occurs at x = x₀ when:

1. The first derivative equals zero: f'(x₀) = 0
2. The second derivative is negative: f''(x₀) < 0

This means:

• The slope of the tangent line at x₀ is zero (tangent is horizontal)
• The function is concave downward at that point
• The function value f(x₀) is greater than values at nearby points

Example: For f(x) = -x² + 4x - 3, the maximum occurs at x = 2 where f'(2) = 0 and f''(2) = -2 < 0

The definite integral $\int_{a}^{b} f(x) dx$ represents the net area between the function $f(x)$ and the x-axis from $x=a$ to $x=b$, where:

• Areas above the x-axis count as positive
• Areas below the x-axis count as negative
• The total area can be approximated by dividing the interval $[a,b]$ into small subintervals of width $\Delta x$ and summing the areas of rectangles
• As $\Delta x$ approaches zero, this approximation approaches the exact area
• Mathematically: $\int_{a}^{b} f(x) dx = \lim_{\Delta x \to 0} \sum_{i=1}^{N} f(x_i)\Delta x$

Example: Finding the area under a velocity-time curve gives the displacement during that time interval.

A Riemann sum approximates the area under a curve by:

1. Dividing the interval $[a,b]$ into $N$ equal subintervals, each with width $\Delta x = \frac{b-a}{N}$
2. Creating rectangles with:
   • Base = $\Delta x$
   • Height = function value $f(x_i)$ at some point in each subinterval
3. Summing the areas of all rectangles: $\sum_{i=1}^{N} f(x_i)\Delta x$

Relationship to definite integrals:

• The definite integral equals the limit of the Riemann sum as $N \to \infty$ (or $\Delta x \to 0$)
• Mathematically: $\int_{a}^{b} f(x) dx = \lim_{N \to \infty} \sum_{i=1}^{N} f(x_i)\Delta x$

Different types of Riemann sums use different evaluation points ($x_i$):

• Left Riemann sum: leftmost point of each subinterval
• Right Riemann sum: rightmost point of each subinterval
• Midpoint Riemann sum: midpoint of each subinterval

Example: Calculating the work done by a variable force requires summing small contributions over distance intervals.

Rules for significant figures in calculations:

For multiplication/division:

• The result has the same number of significant figures as the measurement with the FEWEST significant figures
• Example: 12.0 kg ÷ 7.0 kg = 1.7 (not 1.714285...)
  • Both measurements have 2 significant figures, so result has 2 significant figures

For addition/subtraction:

• The result has the same number of DECIMAL PLACES as the measurement with the fewest decimal places
• Example: 24.36 + 0.0623 + 256.2 = 280.7
  • 256.2 has only 1 decimal place, so result has 1 decimal place

Process for addition/subtraction:

1. Identify the measurement with fewest decimal places
2. Align all numbers by decimal point
3. Round all values to match the measurement with fewest decimal places
4. Perform the calculation
5. Round the final answer to match the fewest decimal places

Note: Remember that digits before the decimal point don't matter for addition/subtraction rules; only the number of decimal places matters.

Measurement uncertainty is expressed as a range around the mean value where the true value likely lies:

• $\bar{x} \pm \sigma$: There's a 68% probability that the true value lies within this interval

  • This is a standard uncertainty interval based on normal distribution

• $\bar{x} \pm 1.96\sigma$: There's a 95% probability that the true value lies within this interval

  • This is commonly used in scientific research as a "95% confidence interval"

• $\bar{x} \pm 3\sigma$: There's a >99% probability that the true value lies within this interval

  • This provides near-certainty and is used in critical applications

Proper reporting format:

• Measurement result = ($\bar{x} \pm \Delta x$) units where $\Delta x$ is typically $\sigma$ or a multiple of $\sigma$

Example: If repeated measurements of focal length yield $\bar{x} = 25.4$ cm with $\sigma = 0.2$ cm, report as: f = (25.4 ± 0.2) cm

Note: These probability estimates are valid when the number of measurements N > 8 and errors follow normal distribution.

The method of components involves:

1. Breaking each vector into its x and y components using trigonometry

   • For a vector of magnitude A at angle θ:
   • x-component = A·cos(θ)
   • y-component = A·sin(θ)

2. Adding all x-components together to get the resultant x-component

3. Adding all y-components together to get the resultant y-component

4. Finding the magnitude using the Pythagorean theorem: R = √(Rx² + Ry²)

5. Finding the direction using the inverse tangent: θ = tan⁻¹(Ry/Rx)

This method is more versatile than the parallelogram method because:

• It works for any number of vectors, not just two
• It provides precise numerical results rather than graphical approximations
• It handles complex angular relationships more easily
• It's systematic and works in any number of dimensions when extended

Example: Adding vectors of 5.0m at 37°, 3.0m along x-axis, and 2.0m along y-axis gives a resultant of 8.6m at 35.5° from the x-axis.

Vector Operations with Equal-Magnitude Vectors

For two vectors $\vec{A}$ and $\vec{B}$ of equal magnitude (5 units) at angle $\theta = 60°$:

1. Resultant Vector Magnitude ($|\vec{A} + \vec{B}|$):

• Formula: $|\vec{A} + \vec{B}| = \sqrt{A^2 + B^2 + 2AB\cos\theta}$
• Calculation:
  • $|\vec{A} + \vec{B}| = \sqrt{5^2 + 5^2 + 2(5)(5)\cos60°}$
  • $= \sqrt{25 + 25 + 50(0.5)}$
  • $= \sqrt{75} = 5\sqrt{3} \approx 8.66$ units

2. Difference Vector Magnitude ($|\vec{A} - \vec{B}|$):

• Formula: $|\vec{A} - \vec{B}| = \sqrt{A^2 + B^2 - 2AB\cos\theta}$
• Calculation:
  • $|\vec{A} - \vec{B}| = \sqrt{5^2 + 5^2 - 2(5)(5)\cos60°}$
  • $= \sqrt{25 + 25 - 25}$
  • $= \sqrt{25} = 5$ units

Key Insight: For equal-magnitude vectors at angle $\theta$:

• When $\theta = 0°$ (parallel): Maximum resultant ($2A$), minimum difference (0)
• When $\theta = 180°$ (antiparallel): Minimum resultant (0), maximum difference ($2A$)
• When $\theta = 60°$: The difference vector has the same magnitude as either original vector

Alternative Approach: The difference vector can also be calculated by considering $\vec{A} + (-\vec{B})$, where the angle between them is $180° - 60° = 120°$.