Adding Vectors Geometrically — Question 9

Question 9

Drawing uncertainty limits what a graphical sum can certify

A head-to-tail drawing uses the exact scale 1.0 m per paper centimetre. The initial tail O is exact. The nominal final head P lies 5.0 cm due east of O. Independent construction checks guarantee only that the true final head lies anywhere in the closed disk of radius 0.20 cm centered at P. This is a hard bound on the completed drawing, not a separate bound for every leg.

1. Find tight lower and upper bounds on the resultant magnitude. Describe endpoint locations attaining them.

2. Can the drawing certify a nonzero eastward resultant? Can it certify that the direction is exactly east? Explain the distinction.

3. Tangents from O to the uncertainty disk bound the possible direction angle. Use right-triangle geometry to find the largest angular deviation from east; you may use inverse sine.

4. Suppose instead that P were only 0.10 cm east of O with the same disk radius. What would change about zero-result feasibility, possible directions and an assigned direction for a zero vector?

Question 9 - Solution

Drawing uncertainty limits what a graphical sum can certify

(a) Use the nearest and farthest points of the endpoint disk.

$$\boxed{R_{\min}=5-.2=4.8\,\mathrm m,\quad R_{\max}=5+.2=5.2\,\mathrm m}$$ The disk points nearest to and farthest from O attain the bounds. The exact calibration converts their paper distances directly to metres.

(b) A directional sector is weaker than an exact heading.

$$x_{\min}=4.8\,\mathrm m>0,\qquad y\in[-.2,.2]\,\mathrm m\quad\text{need not vanish}$$ Every endpoint has a positive east coordinate, certifying a nonzero eastward component. Nonzero north or south offsets remain possible, so exact east is not certified.

(c) The limiting rays touch the disk tangentially.

$$\sin\alpha_{\max}=\frac{.2}{5}=.04\Rightarrow\alpha_{\max}=2.292^\circ$$ Radius PT is perpendicular to the limiting tangent OT. The right triangle uses OP as hypotenuse. The angular extremes occur at different points from the magnitude extremes.

(d) A disk containing the origin changes the inference qualitatively.

$$OP=.1\,\mathrm{cm}<.2\,\mathrm{cm}\Rightarrow O\text{ inside disk},\qquad0\le R\le.3\,\mathrm m$$ Because O lies strictly inside the disk, sufficiently short arrows in every direction are allowed and zero is feasible. Zero itself has undefined direction, distinct from uncertain directions of nonzero candidates.