Application of Vector Question?

Question

The question states:

To a person riding a bike with velocity (-5 j) km/h the wind seems to have velocity (-9 i - 7 j) km/h.

I would like to state the true velocity of the wind in i, j vector form with the velocity expressed in km/h

sorry!, i forgot to add also

What is the speed of the wind? (How do i work out)
speed (km/h)

Taking i as due East, what is the direction of the wind (in standard bearing notation) to the nearest degree?
bearing (°) =

Answer 1

I just ran across this problem. Others have the wind vector figured out ... you add -5j to the vector -9i - 7j to get the answer -9i - 12j. That's the true wind velocity.

But you had two other parts to your question also. To get the wind speed, use the Pythagorean Theorem on that vector. You can do that because the i and j components are perpendicular to each other, so you have right triangles.

Wind speed = sqrt(9^2 + 12^2) = sqrt((81 + 144) = 15 km/h (answer)

You also wanted the wind direction in degrees. This is a bit more complicated. The velocity vector -9i - 12j shows the wind is blowing west and south. One way to get the degrees is to say north is 0 or 360 degrees, east is 90, south is 180, and west is 270.

Here, the wind blows southwest, or west of south. I'll set up x as the angle of the wind from the south. Then tan x = the i (west) component over the j (south) component:

tan x = 9/12 = 3/4 = 0.75

This means x is 36.9 degrees. But that's 36.9 degrees west of south, so to get the final answer, we add 180 (for south). Rounding off to the nearest degree, the wind direction is 180 + 37 = 217 degrees. (Answer)

Answer 2

The wind velocity is -9i-12j.

The i vector is independent of the bike, so it stays the same.

As for the j vector, imagine being on a moving sidewalk (at 5 mph) with a person with a "wind" sign on him walking in the same direction as you at what seems to be 7 mph. He's really walking 12 mph, right? That's the -12j.

Answer 3

The true velocity of the wind is the sum velocity of the wind in the reference frame and the velocity of the reference frame:
-9i-7j kph +(-5j kph)
= -9i-13j kph