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. On a safari, a team of naturalists sets out toward a research station located 9.6 km away in a direction 42° north of east. After traveling in a straight line for 3.1 km, they stop and discover that they have been traveling 25° north of east, because their guide misread his compass. What is the direction (relative to due east) of the displacement vector now required to bring the team to the research station?

Respuesta :

Answer:[tex]\theta =49.76^{\circ}[/tex] North of east

Explanation:

Given

Research station is 9.6 km away in [tex]42^{\circ}[/tex]North of east

after travelling 3.1 km [tex]25^{\circ}[/tex] north of east

Position vector of safari after 3.1 km is

[tex]r_2=3.1cos25\hat{i}+3.1sin25\hat{j}[/tex]

Position vector if had traveled correctly is

[tex]r_0=9.6cos42\hat{i}+9.6sin42\hat{j}[/tex]

Now applying triangle law  of vector addition we can get the required vector[tex](r_1)[/tex]

[tex]r_1+r_2=r_0[/tex]

[tex]r_1=(9.6cos42-3.1cos25)\hat{i}+(9.6sin42-3.1sin25)\hat{j}[/tex]

[tex]r_1=4.325\hat{i}+5.112\hat{j}[/tex]

Direction is given by

[tex]tan\theta =\frac{y}{x}=\frac{5.112}{4.325}[/tex]

[tex]\theta =49.76^{\circ}[/tex]

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