Adding Vectors Geometrically — Question 8

Question 8

A closing vector has the opposite sense to the open-chain sum

A survey chain follows 12 m east, then 5 m north, from starting point O to endpoint P. A third straight displacement $\vec C$ is chosen to return exactly to O. All lengths and directions are exact, and the survey is in one horizontal plane.

1. Find the resultant of the first two legs and the required closing vector C. Include magnitude and direction for each.

2. Write the vector equation for the closed chain. Explain why C is not equal to the first two legs’ sum even though their magnitudes agree.

3. Find total distance for the closed route. Does a zero resultant imply that every arrow must have an equal and opposite partner among the other arrows?

4. A student draws C with the correct 13 m length but in the direction O–P. Find the new full-chain resultant and decide whether the new route closes.

Question 8 - Solution

A closing vector has the opposite sense to the open-chain sum

(a) The open-chain resultant and closure share a line but oppose each other.

$$\boxed{R=\sqrt{12^2+5^2}=13\,\mathrm m},\quad\theta=\tan^{-1}(5/12)=22.620^\circ,\quad\theta_C=202.620^\circ$$ The open-chain sum points northeast. The equal-length closing vector points $22.620^\circ$ south of west, from P back to O.

(b) Closure requires zero directed change.

$$\vec A+\vec B+\vec C=\vec0\Rightarrow\vec C=-(\vec A+\vec B)$$ The closing arrow is opposite to the nonzero sum, so equal lengths do not make the vectors equal.

(c) A polygon can close without pairwise opposite sides.

$$s=12+5+13=30\,\mathrm m,\qquad\Delta\vec r=\vec0$$ The three arrows close collectively; no individual pair has to be equal and opposite. The 30 m distance is a sum of positive lengths.

(d) The incorrectly directed arrow adds another copy of the resultant.

$$\vec R_{\mathrm{wrong}}=\vec R+\vec R=2\vec R=(24,10)\,\mathrm m,\quad R_{\mathrm{wrong}}=26\,\mathrm m$$ Adding an arrow in the open sum’s direction doubles the original error. The resulting chain remains open with the same northeast heading.