What is the orientation of u = (-4, -2)T ?
By plugging into the formula (and using inv tan of the MS Win calculator):
arc tan( y/x ) = arc tan( -2/-4) = arc tan( 0.5 ) = 26.565 degrees
But from the sketch, this is in the wrong quadrent. Study the sketch and use geometry to see that the answer must be (180 + 26.565 )o = 206.565o.
Say that (using a particular coordinate frame) you know that a vector is represented by the column matrix (x, y)T.
You can calculate the two attributes of the vector:
orientation of (x, y)T = arc tan( y/x )
| (x, y)T | =( x2 + y2 )
In a few pages we will start with length and orientation and calculate the x and y for a particular coordinate frame.
Sketch the vector r represented by (4, 5)T. Make an eyeball estimate of its length and direction. Then do the math to get the exact answer.