# Math Help - vectors problem.

1. ## vectors problem.

An ultralight plane si headed N30W at 40km/h. A 12km/h wind is blowing in the direction E20S. What is the resultant velocity of the ultralight plane with respect to the ground?

I cannot figure out how to draw the wind blowing E20S.

2. Is this sketch okay for you to solve the problem?

The longer black arrow is the motion of the plane and the shorter black arrow is the direction of the wind. And red double arrowhead is the resultant motion of the plane.

3. ## no angles

"In mathematics, you don't understand things. You just get used to them." -- Johann von Neumann

I would have wanted to solve the triangle using the cosine/sine law but I can not see an angle. If I go with a scale and 1cm=4km/h I get the resultant 7cm/21km/h
Any suggetions to solve it using trigonometry?

4. method of components ...

$A_x + W_x = G_x$

$-40\sin(30) + 12\cos(20) = G_x$

$A_y + W_y = G_y$

$40\cos(30) - 12\sin(20) = G_y$

$|G| = \sqrt{(G_x)^2 + (G_y)^2}$

$\theta = \arctan\left(\frac{G_y}{G_x}\right) + 180$

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6. And you can still use geometry:

Put another direction at the upper tip of the triangle. From parallel lines, you know that the lower angle is 30 degrees. And since those angles are found in a right angle, you can deduce that the middle angle is 90 - (30 + 20) = 40 degrees.

From there, you can use the cosine rule and the sine rule to get angle x, which you use to determine the resultant angle.