# Vector applications help

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• Jun 17th 2010, 01:52 PM
john23
Vector applications help

A plane is supposed to travel 820km on a bearing of 80 degrees. The plane's speed is 800km/h. The wind is blowing at 70km/h on a bearing of 100 degrees. What heading should the plane take and how long will the trip take?
• Jun 17th 2010, 04:54 PM
Ackbeet
What work have you done so far?
• Jun 17th 2010, 05:15 PM
john23
This is what I have done so far (the arrows are meant to be on top of the letter to indicate a vector):

Let ->V be the wind
Let ->U be the plane’s speed
Let ->W be the plane’s heading

Vx = 70cos10°
Vy = 70sin10°

Ux = 800cosθ
Uy = 800sinθ

but I don't know how to find the resultant vector without theta.
• Jun 17th 2010, 05:21 PM
Ackbeet
I'm assuming you meant

Vx = 70 cos(100°)
Vy = 70 sin(100°), right?

Don't you already know the desired resultant vector? Perhaps you could write out an equation that would help you find what you need...
• Jun 17th 2010, 05:27 PM
john23
That is what I meant.

I dont know the desired resultant vector, only the total distance not the speed. I need to find the resultant vector of the 800km/h speed and the 70km/h wind, but I don't know how to do this since i dont know the angle of the speed, I only know the angle of the resultant vector which is 10 degrees.
• Jun 17th 2010, 05:29 PM
Ackbeet
Is the plane's speed 800 km/hr in air, or relative to the ground? I suppose I'm also asking this: what are the assumptions regarding how the velocities combine? Do they just add vectorially?
• Jun 17th 2010, 05:34 PM
john23
How the speed is measured is irrelevant, they just add vectorially.
• Jun 17th 2010, 05:38 PM
Ackbeet
Ok. Suppose you do this:

800 cos(θ) + 70 cos(100°) = S cos(80°)
800 sin(θ) + 70 sin(100°) = S sin(80°),

where S is the unknown resultant speed. Now, it seems to me that you have here two equations in two unknowns. How could you go about solving this?
• Jun 17th 2010, 06:08 PM
john23
Oh!!! That makes sense, I don't know why I didn't think of that. Ok so now I have to use either substitution or elimination to solve.
• Jun 17th 2010, 06:09 PM
Ackbeet
Yeah, you could do that. Except that one of your variables does not appear in the equations in a linear fashion (which, I think, rules out elimination). I think the first thing I would do is divide one equation by the other to eliminate S.
• Jun 17th 2010, 06:25 PM
john23
Once I divide the equations how to I solve for theta since there are two of them.
• Jun 17th 2010, 06:29 PM
Ackbeet
Just take it one step at a time. What do you get for the division? And what do you suppose the next step after that would be?
• Jun 17th 2010, 06:39 PM
john23
I get:
800sin(θ) - 70sin(10) = sin(10)
800cos(θ) + 70cos(10) = cos(10)

I think the next step would be:

800tan(θ) - 0.18 = 0.176
θ = tan-1 ( (0.176 + 0.18) / 800)

Is that right?
• Jun 18th 2010, 02:30 AM
Ackbeet
Addition does not distribute over division. Your second step is therefore incorrect. You've got:

800 sin(θ) + 70 sin(10)
--------------------------- = tan(10),
800 cos(θ) + 70 cos(10)

Now you must multiply both sides by the denominator of the LHS:

800 sin(θ) + 70 sin(10) = tan(10) (800 cos(θ) + 70 cos(10)).

Get all the terms with θ over to the LHS, and all terms without θ over to the RHS.

One recommendation: don't substitute in numbers for anything until the very end. Why? Professors love to give you similar problems with slightly different initial conditions. If you've algebraically solved for the answer, you can just plug those new conditions in to your final answer. If you've plugged in numbers too early, you'll have to re-do everything. It's more work!
• Jun 18th 2010, 02:54 AM
john23
Once I have all the terms with theta to one side how do I solve for both?
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