ANSWER
46 miles/hr with a direction angle of 32°
EXPLANATION
Given:
• The velocity of the plane, ,v, = 70 mi/h at 24°
,• The velocity of the wind, ,u, = 25 m/h at 190°
Find:
• The resultant velocity of the plane, ,u, + ,v
First, let's draw each vector,
The resultant is the sum of the two vectors, so first, we have to find the vectors in the form . The components of a vector form a right angle with the vector itself. The x-component is the adjacent side to the direction's angle and the y-component is the opposite side.
For the velocity of the car, we have,
[tex]\vec{v}=<70\cos24\degree,70\sin24\degree>[/tex]The two components of the wind's velocity are negative because the vector is on the third quadrant. The angle in the triangle is 190 - 180 = 10°,
[tex]\vec{u}=\lt-25\cos10\degree,-25\sin10\degree>[/tex]The sum of the two vectors is,
[tex]\vec{v}+\vec{u}=\lt(70\cos24\degree-25\cos10\degree),(70\sin24\degree-25\sin10\degree)>[/tex]Let's solve this,
[tex]\vec{v}+\vec{u}=\lt39.3280,24.1304>[/tex]The magnitude of the resultant, by the Pythagorean Theorem, is,
[tex]||\vec{v}+\vec{u}||=\sqrt{39.3280^2+24.1304^2}\approx46.1407\approx46\text{ }mi/h[/tex]And using the tangent of the direction angle, we can find the direction of the resultant,
[tex]\theta=\tan^{-1}\left(\frac{24.1304}{39.3280}\right)\approx31.5319\degree\approx32\degree[/tex]Hence, the resultant is 46 miles/hr with a direction angle of 32°, rounded to the nearest whole number.
