Question 327541
A vector is described. Express the vector in terms of i and j. If exact values are not possible, round components to the nearest tenth. 

A plane with an airspeed of 450 mph is flying in the direction N35W.

I have gotten an answer but am not sure if I have done it right. I need some help on this to make sure. Are you to use bearings?
<pre>
North is the positive y-axis
South is the negative y-axis
East is the positive x-axis
West is the negative x-axis

This is the vector.  Its direction is determined by starting from 
north, the positive y-axis, and swinging a vector toward the west,
which is counter-clockwise, by 35°. 

{{{
drawing(400,400,-1,1,-1,1, red(line(0,0,-cos(55*pi/180),sin(55*pi/180))),
line(-1,0,1,0), line(0,-1,0,1), locate(0.05,1,N),
locate(.9,0.1,E),locate(0.05,-.9,S),locate(-.9,0.1,W),
red(line(-.43,.71,-cos(55*pi/180),sin(55*pi/180))), 
arc(0,0,.4,-.4,90,125), locate(-.13,.3,"35°"),
red(line(-.53,.65,-cos(55*pi/180),sin(55*pi/180))),

 locate(-0.1,1,"+y"),
locate(.95,-0.01,"+x"),locate(-0.13,-.91,-y),locate(-.97,-0.01,-x)







)}}}

To express this vector in terms of horizontal and vertical
component vectors, we draw horizontal and vertical lines from the
tip of that vector to the x and y axes:

{{{
drawing(400,400,-1,1,-1,1, red(line(0,0,-cos(55*pi/180),sin(55*pi/180))),
line(-1,0,1,0), line(0,-1,0,1), locate(0.05,1,N),
locate(.9,0.1,E),locate(0.05,-.9,S),locate(-.9,0.1,W),
red(line(-.43,.71,-cos(55*pi/180),sin(55*pi/180))), 
arc(0,0,.4,-.4,90,125), locate(-.13,.3,"35°"),
red(line(-.53,.65,-cos(55*pi/180),sin(55*pi/180))),
rectangle(-cos(55*pi/180),0,0,sin(55*pi/180),0),
 locate(-0.1,1,"+y"),
locate(.95,-0.01,"+x"),locate(-0.13,-.91,-y),locate(-.97,-0.01,-x)


)}}}

Then these two green vectors are the x-component
and y-component of the red vector:

{{{
drawing(400,400,-1,1,-1,1, red(line(0,0,-cos(55*pi/180),sin(55*pi/180))),
green(line(0,0,-cos(55*pi/180),0)), line(-cos(55*pi/180),0,-1,0),

 locate(-0.1,1,"+y"),
locate(.95,-0.01,"+x"),locate(-0.13,-.91,-y),locate(-.97,-0.01,-x),

line(0,0,1,0), 




line(0,-1,0,1), locate(0.05,1,N),
locate(.9,0.1,E),locate(0.05,-.9,S),locate(-.9,0.1,W),
red(line(-.43,.71,-cos(55*pi/180),sin(55*pi/180))), 
arc(0,0,.4,-.4,90,125), locate(-.13,.3,"35°"),
red(line(-.53,.65,-cos(55*pi/180),sin(55*pi/180))),
rectangle(-cos(55*pi/180),0,0,sin(55*pi/180),0),
green(line(-.4,.07,-cos(55*pi/180),0)),
 locate(-0.1,1,"+y"),
locate(.95,-0.01,"+x"),locate(-0.13,-.91,-y),locate(-.97,-0.01,-x),
green(line(-.4,-.07,-cos(55*pi/180),0)),
green(line(-.4,.07,-cos(55*pi/180),0)),
green(line(0,0,0,sin(55*pi/180))),
green(line(-.07,.68,0,sin(55*pi/180))),
green(line(.07,.68,0,sin(55*pi/180)))
)}}}

The actual angle is this vector measured from the 
postive x axis, indicated by this blue arc, which is
90°+35° or 125°

{{{
drawing(400,400,-1,1,-1,1, red(line(0,0,-cos(55*pi/180),sin(55*pi/180))),
green(line(0,0,-cos(55*pi/180),0)), line(-cos(55*pi/180),0,-1,0),
 locate(-0.1,1,"+y"), locate(.2,.4,"125°"),
locate(.95,-0.01,"+x"),locate(-0.13,-.91,-y),locate(-.97,-0.01,-x),
blue(arc(0,0,.8,-.8,0,125)),

line(0,0,1,0), 




line(0,-1,0,1), locate(0.05,1,N),
locate(.9,0.1,E),locate(0.05,-.9,S),locate(-.9,0.1,W),
red(line(-.43,.71,-cos(55*pi/180),sin(55*pi/180))), 
arc(0,0,.4,-.4,90,125), locate(-.13,.3,"35°"),
red(line(-.53,.65,-cos(55*pi/180),sin(55*pi/180))),
rectangle(-cos(55*pi/180),0,0,sin(55*pi/180),0),
green(line(-.4,.07,-cos(55*pi/180),0)),

green(line(-.4,-.07,-cos(55*pi/180),0)),
green(line(-.4,.07,-cos(55*pi/180),0)),
green(line(0,0,0,sin(55*pi/180))),
green(line(-.07,.68,0,sin(55*pi/180))),
green(line(.07,.68,0,sin(55*pi/180)))
)}}}

The magnitude of the green horizontal vector is {{{450cos("125°")}}}

Therefore it is the vector {{{450cos("125°")}}}<i><b>i</b></i>

The magnitude of the green vertical vector is {{{450sin("125°")}}}

Therefore it is the vector {{{450sin("125°")}}}<i><b>j</b></i>

So to write the red vector in terms of the horizontal unit vector
 <i><b>i</b></i> and the vertival unit vector <i><b>j</b></i>, we have

{{{450cos("125°")}}}<i><b>i</b></i> + {{{450sin("125°")}}}<i><b>j</b></i>

There is no way to get the exact values, so we punch those into a 
calculator and get:

{{{-258.1093964}}}<i><b>i</b></i> + {{{368.6184199}}}<i><b>j</b></i>

Or rounded to tenths:

{{{-258.1}}}<i><b>i</b></i> + {{{368.6}}}<i><b>j</b></i>

Edwin</pre>