A plane flies 45km in a direction 35 degree south of east, then turns and flies in a direction 75 degree north of east for 65km. Find the final position of the plane. Solve this problem using the...

A plane flies 45km in a direction 35 degree south of east, then turns and flies in a direction 75 degree north of east for 65km. Find the final position of the plane. Solve this problem using the parallelogram method of vector addition. Be sure to give the final position in terms of the distance and direction of the plane from its starting point.

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Lix Lemjay eNotes educator | Certified Educator

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To solve using parallelogram method, draw the two vectors on the same initial point. (See Fig.1 in the attachment.)

Then, draw another two lines in such a way that a parallelogram is form. (See Fig.2)

The angle between the two vectors is:

`35^o +75^o=110^o`

Applying the property of a parallelogram --- consecutive angles are supplementary, then the other angle is:

`180^o - 110^o=70^o`

And draw a diagonal starting from the initial position. This represents the resultant vector. (See Fig.3)

To solve for the magnitude of the resultant vector, apply Cosine Law.

`c^2=a^2+b^2-2abcosC`

`R^2=45^2+65^2-2(45)(65)cos(70^o)`

`R^2=4249.18`

`R=65.19 km`

To solve for theta, apply Cosine Law again.

`b^2=a^2+c^2-2ac cosB`

`65^2=45^2+65.19^2-2*45*65.19cos(theta)`

`cos(theta)= (45^2+65.19^2-65^2)/(-2*45*65.19)`

`theta = cos^(-1)((45^2+65.19^2-65^2)/(-2*45*65.19))`

`theta=69.55^o`

Using the East axis as the reference for its direction, then the direction of the resultant vector is:

`theta_f = 69.55^o - 35^o`

`theta_f =34.55^o` (See Fig.4)

Therefore, the final position of the plane is 65.19km away from its starting point and its direction is 34.55 degree North of East.

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