Question 1161805
{{{drawing(340,340,-210,130,-210,130,
grid(0),line(0,0.5,-190,0.5),line(0,-0.5,-190,-0.5),
line(0,0,-190,0),arrow(0,0,-190,0),
arrow(-190,0,-41.51,-148.49),
arrow(-41.51,-148.49,93.49,85.33),
red(arrow(93.49,85.33,0,0)),
green(arrow(-190,10,-190,-100)),
green(arrow(-41.51,-160,-41.51,-50)),
green(arrow(93.47,85.34,93.47,-20)),
locate(-190,-101,green(S)),
locate(-41.51,-35,green(N)),
locate(93.47,-21,green(S)),
green(arc(-190,0,80,80,45,90)),
green(arc(93.47,85.34,80,80,90,135)),
green(arc(-41.51,-148.47,80,80,-90,-60)),
locate(-192,20,A),locate(-41,-148,C),locate(93.5,100,D)
)}}} We start from point {{{O(0,0)}}} ,
moving 190 units West along the negative x-axis to point {{{B}}}
The components of vector {{{OB}}} are
{{{OB[x]=-190}}} and {{{OB[y]=0}}} .
We reach {{{B(-190,0)}}}
From there we go 210 units to point {{{C}}} in a direction {{{45^o}}} to the East of {{{green(S)}}} .
That is {{{45^o+270^o=315^o}}} counterclockwise from the positive x-axis.
{{{BC[x]=210*cos(315^o)=148.49}}} and {{{BC[y]=210*cos(315^o)=-148.49}}} .
The coordinates of {{{C}}} are
{{{x[C]=-190+148.49=-41.51}}} and {{{y[C]=0+(-148.49)=-148.49}}}
Then we head to point {{{D}}} for {{{270}}} units
in a direction that is {{{30^o}}} to the East of {{{green(N)}}} .
That is {{{90^o-30^o=60^o}}}  counterclockwise from the positive x-axis.
{{{CD[x]=270*cos(60^o)=135}}} and {{{CD[y]=270*sin(60^o)=233.83}}} .
The coordinates of {{{C}}} are
{{{x[D]=-41.51+135=93.49}}} and {{{y[D]=-148.49+233.83=85.33}}} .
Finally we head back to starting point {{{O(0,0)}}} .
The distance of that fourth displacement is
{{{abs(OD)=sqrt(93.49^2+85.33^2)=highlight(126.58)}}}
The angle {{{alpha}}} that {{{OD}}} makes with the positive x-axis is such that
{{{tan(alpha)=95.33/93.49=0.9127}}} and {{{alpha=42.39^o}}}
The angle {{{theta}}} the direction {{{DO}}} makes with the direction {{{green(S)}}} is
{{{90^o-42.39^o=highlight(47.61^o)}}}
and the direction {{{DO}}} is to the West of {{{green(S)}}} .
It forms a counterclockwise angle of {{{270^o-47.61^o=222.39^o}}} with the positive x-axis
For verification:
The components of vector {{{DO}}} are
{{{DO[x]=0-93.5=-93.5=126.58*cos(222.39^o)}}} and
{{{DO[y]=0-85.3=-85.3=126.58*sin(222.39^o)}}}