Expert: Scott A Wilson - 2/7/2012

Question
Scott,
I'm doing this class online and I really just want to graduate so far I've been doing good until now...I don't get this and it confuses me. If you could be kind enough to solve these and explain how to do them it would save me and I would greatly appreciate it. I just need some examples to go off of, you know? These online classes never explain anything perfectly so I'm looking for a real live person to try and help me out. Thank you, very, very much.


These are 10 examples and I want to see if you can explain them to me. My teacher isn't the best around and I'm just looking for some help and examples I can go off.

The magnitude and direction of vectors u and v are given. Find vector w's polar coordinates.
u: magnitude 140, bearing 160°
v: magnitude 200, bearing 290°
w: u + v

(A) w: (282, 141°)
(b)w: (23603, 226°)

(c)w: (153.6, 246°)
(d)w: (22, 182°)

2. The magnitude and direction of vectors u and v are given. Find vector w's polar coordinates.
u: magnitude 30, bearing 215°
v: magnitude 30, bearing 110°
w: 3u - v
(a)w: (17, 149°)  (b)w: (102, 231.5°)
(c)w: (193, 232°)  (d)w: (29, 185°)

3. The magnitude and direction of vectors u and v are given. Find vector w's polar coordinates.
u: magnitude 10, bearing 30°
v: magnitude 25, bearing 120°
w: 8u + 3v
(a)w: (99, 56°)  (b)w: (110, 73°)
(c)w: (312, 63°)  (d)w: (890, 42°)

4.
The magnitude and direction of vectors u and v are given. Find vector w's polar coordinates.
u:  magnitude 15, bearing 60°
v:  magnitude 20, bearing 160°
w:  2u - 3v
(a)w: (35, 110°)  (b)w: (71.6, 4.5°)
(c)w: (4.5, 71.6°)  (d)w: (52.0, 260°)

5. The magnitude and direction of vectors u and v are given. Find vector w's polar coordinates.
u: magnitude 136, bearing 220°
v: magnitude 197, bearing 300°
w: u + v
(a)w: (258.1, 268.7°)  (b)w: (268.7, 258.1°)
(c)w: (55.2, 223.4°)   (d)w: (223.4, 55.2°)

6. The magnitude and direction of vectors u and v are given. Find vector w's polar coordinates.
u: magnitude 218, bearing 22°
v: magnitude 170, bearing 112°
w: u - v
a()  (133.2, 72°)  b(72, 133.2°)
c(276, 344.1°)  d(344.1, 276°)

7. The magnitude and direction of vectors u and v are given. Find vector w's polar coordinates.
u: magnitude 136,. bearing 220°
v: magnitude 197, bearing 300°
w: 2u - v  a(223, 104°)  b(104, 223°)
c(180.8, 307°)  d(307, 180.8°)

8. The magnitude and direction of vectors u and v are given. Find vector w's polar coordinates.
u: magnitude 1850, bearing 125°
v: magnitude 2960, bearing 25°
w: u + 2v  a(5888, 43.0°)  b(43, 5888.0°)
c(345, 22.2°)  d(22, 354.2°)

9. The magnitude and direction of vectors u and v are given. Find vector w's polar coordinates.
u: magnitude 460, bearing 0°
v: magnitude 712, bearing 130°
w: u + v  a(90, 545°)  b(545, 90°)
c(220, 30°)  d(30, 220°)

10. The magnitude and direction of vectors u and v are given. Find vector w's polar coordinates.
u: magnitde 23, bearing 215°
v: magnitude 14.5, bearing 105°
w: u + v  a(22.7, 77°)  b(77, 22.7°)
c(22.6, 177.9°)  d(177.9, 22.6°)

Answer
To solve these, I will call the distances u and v.
In the x direction, I will call the distance ux and vx.
In the y direction, I will call the distance uy and vy.
The angles for u and v will be called uA and vA, respectively.

It is known that ux/u = cos(uA), uy/u = sin(uA), vx/v = cos(vA), and vy/v = sin(vA).
These can be rewritten as ux = u*cos(uA), uy = u*sin(uA), vx = v*cos(vA), and vy = v*sin(vA).

The x distance is then ux + vx.
The y distance is then vx + vy.

When multiples of the directions u and v are taken,
multiply the original distance by that much before
finding the u distance and the v distance for each angle.
The measurements in the horizontal and vertical distances are ux, uy, vx, and vy.

The total magnitude of wx is ux+vx.
The total magnitude of wy is uy+vy.
Tht magitude of w is √(wx²+wy²).
If we take wy/wx, we get the tangent of the beariing of the sum of the two movements.


For an example, lets look at 4.  It says

The magnitude and direction of vectors u and v are given.
Find vector w's polar coordinates.
u:  magnitude 15, bearing 60°
v:  magnitude 20, bearing 160°
w:  2u - 3v
(a)w: (35, 110°)  (b)w: (71.6, 4.5°)
(c)w: (4.5, 71.6°)  (d)w: (52.0, 260°)

It is known ux = 15*cos(60°), uy = 15*sin(60°), vx = 20*cos(160°), and vy = 20*sin(160°).
That is, ux = 7.5, uy = 12.99, vx = -18.79, vy = 6.84
Since wx = 2*ux - 3*vx = 71.38.  Since wy = 2*uy - 3*vy, wy = 5.46.

To find w, use w = √(wx²+wy²) = 71.6.
To find the angle, it is known that tangent of the angle is tan(w) = wy/wx.
To convert to degrees, it is w*180/pi.