The route followed by a hiker consists of three displacement vectors A, B, and C

Question

The route followed by a hiker consists of three displacement vectors A, B, and C. Vector A is along a measured trail and is 2470 m in a direction 43.0 ° north of east. Vector B is not along a measured trail, but the hiker uses a compass and knows that the direction is 20.0 ° east of south. Similarly, the direction of vector C is 17.0 ° north of west. The hiker ends up back where she started, so the resultant displacement is zero, or A+ B+ C= 0. Find the magnitudes of (a) vector B and (b) vector C .

Explanation / Answer

I'm using positive x axis as east, negative x-axis as west, positive y-axis as north, negative y-axis as south.

43 north of east is thus: + 43 degrees with the positive x-axis:

20 east of south, means 20 degrees to the right of 270 degree angle.

17 degrees north of west means 17 degrees above 180 degrees.

A = 2470(cos(43)i + sin(43)j)

B = K(cos(270 + 20)i + sin(270+20)j )

C = P(cos(180 - 17) i+ sin(180 - 17)j)

For the resultant vector to be 0, the individual components must also be 0:

A + B + C = [2470cos(43) + Kcos(290) + Pcos(163)]i

+ [2470sin(43) + Ksin(290) + Psin(163)]j = 0i + 0j

I'm looking for P and K

Pcos(163) + Kcos(290) = -2470cos(43) (1)
Psin(163) + Ksin(290) = -2470sin(43) (2)

This is a system of equations with two unknowns and two equations. I solved by using a computer algebra system, i.e. using matrix echelon form and I got:

P = 3099 m

K = 2,749 m

This might not be right as I used a couple estimates, but The system of equations should be right though.

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