Vector Addition with Scale Factor: Tip-to-Tail Method

Setup and scale factor

• The setup uses blocks to visually represent length and a scale factor to translate blocks into units.
• Scale factor given: 1 unit = 5 blocks.
• Vectors:
  •  Vector \vec{a} is 2 units long and at angle 60° from the +x axis.
  •  Vector \vec{b} is 1 unit long and at angle 30° from the +x axis (drawn starting at the tip of \vec{a}).
• Practical note: The blocks are used as a measurement aid; the units can be anything (e.g., blocks, pounds) depending on context.
• The end goal: Find the resultant vector \vec{r} = \vec{a} + \vec{b}, and determine its magnitude and direction relative to the starting point.
• A reminder that the scale factor converts measurements in blocks to units, not a simple scalar addition.

Vectors and their representations

• Representation of a vector with direction: a directed line with an arrowhead.
• Vector a:
  • Magnitude: \( |\vec{a}| = 2 \text{ units} \).
  • Direction: (60^{\circ}) from the +x axis.
  • In blocks, length is: (2 \text{ units} \times 5 \text{ blocks/unit} = 10 \text{ blocks}) along the 60° direction.
• Vector b:
  • Magnitude: \( |\vec{b}| = 1 \text{ unit} \).
  • Direction: (30^{\circ}) relative to the +x axis (measured starting at the end of a).
  • In blocks, length is: (1 \text{ unit} \times 5 \text{ blocks/unit} = 5 \text{ blocks}) along the 30° direction.
• Tip-to-tail method (vector addition):
  • Place the tail of \vec{b} at the tip of \vec{a}.
  • The resultant vector \vec{r} is drawn from the origin (tail of \vec{a}) to the tip of \vec{b} (end of the chain).
  • This method is valid for any type of vectors.
• Key caveats mentioned:
  • It’s easy to misplace the second vector if the local coordinate system is not set correctly at the end of the first vector.
  • The angle measurement can be imprecise when done visually with a protractor or by counting blocks.
  • Counting blocks to measure magnitude yields an approximate length in blocks, which then must be converted to units via the scale factor.

Constructing the resultant r (block-based construction)

• Steps in the video:
  • Set up the origin and draw a along the 60° direction for length 10 blocks (since 2 units × 5 blocks/unit).
  • From the end of a, draw b along the 30° direction for 5 blocks (since 1 unit × 5 blocks/unit).
  • The endpoint of b gives the end point of r (the resultant from the original origin).
• The direct measurement approach used in the video:
  • The length of r in blocks was counted as approximately 14.7 blocks.
  • This value is then converted to units using the scale factor: ( \text{blocks} \to \text{units} ) via (1 \text{unit} = 5 \text{blocks}).
  • Calculation: \( |\vec{r}|{blocks} \approx 14.7 \text{ blocks} \Rightarrow \ |\vec{r}|{units} \approx \frac{14.7}{5} = 2.94 \text{ units}. )
  • The instructor notes this as approximately (2.9) units due to rounding and measurement precision.
• Important conceptual note:
  • The magnitude of the resultant is not obtained by simply adding the magnitudes of the two vectors (2 + 1); directions matter.

Magnitude and angle of r (unit-based calculation and results)

• Direct measurement (block-based):
  • Magnitude in blocks: approximately (14.7) blocks.
  • Convert to units using the scale factor: $|\vec{r}| = \frac{14.7\text{ blocks}}{5\text{ blocks/unit}} \approx 2.94\text{ units}.$
  • The video rounds to approximately (2.9) units.
• Direction (angle) measurement:
  • After locating the end point of r, measure the angle with a protractor relative to the starting point (the origin).
  • The video notes the angle is very close to (50^{\circ}), i.e., roughly (50^{\circ}) north of east.
  • Expression in map terms: (2.9\text{ units at }50^{\circ}) or (50^{\circ}\text{ north of east}.
• Observed limitation:
  • The method relies on visual measurement, giving limited precision (described as limited by the protractor/measurement).
• Alternative phrasing mentioned:
  • The angle could be stated as (50^{\circ}) from the +x axis (east) toward the +y direction (north).

Exact component-based calculation (for clarity and accuracy)

• Express each vector in components (in units):
  • • $|\vec{a}| = 2, \quad \alpha = 60^{\circ}.$
  • $\vec{a}_x = |\vec{a}| \cos \alpha = 2 \cos 60^{\circ} = 1,$
  • $\vec{a}_y = |\vec{a}| \sin \alpha = 2 \sin 60^{\circ} = \sqrt{3}.$
  • • $|\vec{b}| = 1, \quad \beta = 30^{\circ}.$
  • $\vec{b}_x = |\vec{b}| \cos \beta = \cos 30^{\circ} = \frac{\sqrt{3}}{2},$
  • $\vec{b}_y = |\vec{b}| \sin \beta = \sin 30^{\circ} = \frac{1}{2}.$
• Sum the components to get the resultant components:
  • $\vec{r}<em>x = \vec{a}</em>x + \vec{b}_x = 1 + \frac{\sqrt{3}}{2},$
  • $\vec{r}<em>y = \vec{a}</em>y + \vec{b}_y = \sqrt{3} + \frac{1}{2}.$
• Magnitude of r (exact form and numerical):
  • $|\vec{r}| = \sqrt{(\vec{r}<em>x)^2 + (\vec{r}</em>y)^2} = \sqrt{\left(1 + \frac{\sqrt{3}}{2}\right)^2 + \left(\sqrt{3} + \frac{1}{2}\right)^2} \approx 2.909.$
• Direction of r:
  • $\theta = \tan^{-1}\left( \frac{\vec{r}<em>y}{\vec{r}</em>x} \right) = \tan^{-1}\left( \frac{\sqrt{3} + \tfrac{1}{2}}{1 + \tfrac{\sqrt{3}}{2}} \right) \approx 50.2^{\circ}.$
• Units and blocks cross-check:
  • In blocks, the magnitude is approximately $|\vec{r}|<em>{blocks} \approx |\vec{r}|</em>{units} \times 5 \Rightarrow |\vec{r}|_{blocks} \approx 2.909 \times 5 \approx 14.545\text{ blocks},$ which aligns with the measured ~14.7 blocks.

Interpretation and connections

• Relationship to prior concepts:
  • Demonstrates the tip-to-tail method to add vectors in the plane.
  • Shows how to convert between different unit representations (blocks vs units) using a scale factor.
  • Reinforces that vector addition is not simply adding magnitudes; directions and components determine the resultant.
• Real-world relevance:
  • Expressing direction as an angle relative to a known axis, e.g., "50° north of east," is common in navigation and physics.
• Alternative representations:
  • The resultant can be described as a magnitude and direction: approximately $|\vec{r}| \approx 2.90\text{ units}, \quad \theta \approx 50.2^{\circ}.$
  • In map terms: about $2.9\text{ units at }50^{\circ},\text{ i.e., }50^{\circ}\text{ north of east}.$
• Practical notes on precision and method:
  • Visual/measurement methods yield approximations (e.g., ~14.7 blocks, ~2.9 units, ~50°).
  • A calculation-based approach using trigonometric components provides a precise result (2.909 units, 50.2°).
• Final takeaway:
  • The scale factor bridges the measured blocks to the desired units.
  • The tip-to-tail approach, combined with component analysis, yields the magnitude and direction of the resultant vector in a consistent way.

Quick reference formulas (specific to this example)

• Scale factor: $1\text{ unit} = 5\text{ blocks}.$
• Vector magnitudes and directions:
  • $|\vec{a}| = 2, \quad \alpha = 60^{\circ}.$
  • $|\vec{b}| = 1, \quad \beta = 30^{\circ}.$
• Components (in units):
  • $\vec{a}<em>x = 2\cos 60^{\circ} = 1, \quad \vec{a}</em>y = 2\sin 60^{\circ} = \sqrt{3}.$
  • $\vec{b}<em>x = \cos 30^{\circ} = \frac{\sqrt{3}}{2}, \quad \vec{b}</em>y = \sin 30^{\circ} = \frac{1}{2}.$
• Resultant components:
  • $\vec{r}<em>x = 1 + \frac{\sqrt{3}}{2}, \quad \\vec{r}</em>y = \sqrt{3} + \frac{1}{2}.$
• Resultant magnitude and angle:
  • $|\vec{r}| = \sqrt{\left(1+\frac{\sqrt{3}}{2}\right)^2 + \left(\sqrt{3}+\frac{1}{2}\right)^2} \approx 2.909.$
  • $\theta = \tan^{-1}\left(\frac{\vec{r}<em>y}{\vec{r}</em>x}\right) \approx 50.2^{\circ}.$
• Block-to-unit conversion for the measured magnitude:
  • $|\vec{r}|<em>{blocks} \approx 14.7 \text{ blocks} \Rightarrow |\vec{r}|</em>{units} = \frac{14.7}{5} \approx 2.94 \text{ units}.$