Adding Vectors by Components — Question 8

Question 8

Bound the magnitude of an uncertain sum

Independent hard component bounds give $\vec A$ with $A_x\in[2,3]$ m, $A_y\in[-1,1]$ m, and $\vec B$ with $B_x\in[-1,0]$ m, $B_y\in[2,3]$ m. Every combination within those intervals is allowed.

1. Find the exact allowed rectangle for $\vec R=\vec A+\vec B$.

2. Find the tight minimum and maximum resultant magnitudes.

3. Find tight direction-angle bounds and explain why the vectors giving extreme angles differ from those giving extreme magnitudes.

Question 8 - Solution

Bound the magnitude of an uncertain sum

(a) Add interval endpoints component by component.

$$R_x\in[2-1,3+0]=[1,3],\quad R_y\in[-1+2,1+3]=[1,4]\quad\mathrm m$$ $R_x\in[1,3]$ m and $R_y\in[1,4]$ m. Because all four input component choices are independent, every point of this rectangle is attainable. In particular the resultant always lies in quadrant I.

(b) Choose the nearest and farthest points from the origin.

$$\boxed{R_{\min}=\sqrt{1^2+1^2}=\sqrt2\,\mathrm m,\quad R_{\max}=\sqrt{3^2+4^2}=5\,\mathrm m}$$ Both coordinates are positive, so the minimum is at $(1,1)$ m and the maximum at $(3,4)$ m. Hence $\sqrt2\leq R\leq5$ m. These bounds are attained, not just conservative estimates.

(c) Optimize the component ratio.

$$\theta_{\min}=\tan^{-1}(1/3)=18.435^\circ,\quad\theta_{\max}=\tan^{-1}(4/1)=75.964^\circ$$ The angle is $\tan^{-1}(R_y/R_x)$. Its minimum is $\tan^{-1}(1/3)\approx18.435^\circ$ at $(3,1)$, and its maximum is $\tan^{-1}(4)\approx75.964^\circ$ at $(1,4)$. Length depends on squared components; angle depends on their ratio, so their optimizing corners differ.