Adding Vectors Geometrically — Question 4

Question 4

Symmetry fixes a sum direction before a ruler is used

Two displacement vectors each have magnitude 10 m. Drawn tail-to-tail, their directions are $60^\circ$ apart and symmetric about east: one points $30^\circ$ north of east and the other $30^\circ$ south of east.

1. Use reflection symmetry and the parallelogram construction to determine the sum direction.

2. Find the magnitude using the congruent right triangles formed by the rhombus diagonals. You may use $\cos30^\circ=\sqrt3/2$.

3. When one arrow is translated to form a head-to-tail triangle, what is the triangle’s interior angle at the join? Explain why it is not $60^\circ$.

4. Keeping both magnitudes 10 m, let the tail-to-tail angle grow from $0^\circ$ toward $180^\circ$, symmetrically about east. Describe how the sum changes, including the direction at the limiting zero result.

Question 4 - Solution

Symmetry fixes a sum direction before a ruler is used

(a) The sum diagonal lies on the symmetry line.

$$R_y=10\sin30^\circ-10\sin30^\circ=0,\quad R_x>0\Rightarrow\text{east}$$ Reflection about east exchanges the two inputs and leaves their sum unchanged. Their nonzero sum therefore lies on the eastward symmetry line.

(b) Bisect the rhombus into right triangles.

$$\boxed{R=2(10)\cos30^\circ=10\sqrt3\,\mathrm m=17.321\,\mathrm m}$$ Bisection of the rhombus gives two congruent right triangles. Twice the adjacent leg is the sum diagonal, whereas 20 m is the two-leg route length.

(c) At the join, one triangle side points backward along the first leg.

$$\psi=180^\circ-60^\circ=120^\circ,\quad R^2=10^2+10^2-2(10)^2\cos120^\circ=300\,\mathrm{m^2}$$ At the join, the triangle side points backward along the first arrow. Its angle with the second arrow is supplementary to the tail-to-tail angle.

(d) Larger angular separation produces more cancellation.

$$R(\phi)=20\cos(\phi/2)\,\mathrm m:\quad R(0)=20\,\mathrm m,\quad R(180^\circ)=0$$ For separations less than $180^\circ$, the resultant points east and decreases continuously in magnitude. At $180^\circ$ it vanishes and its direction is undefined.