Determine the magnitude and direction of the displacement if a manwalks 30.5 km

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Explanation / Answer

We know he has a displacement of 30.5km directly NE, but we have tocome up with a vector... i + j is in the direction of NE,you can see this by Pythagorean's Theorem (i, j, k are thex, y, z direction unit vectors, respectively). However, it is not aunit vector so we need to make it into one...we don't want thevector we choose to change the displacement of the vector we arelooking for. Note: unit vectors have a magnitude of one by usingPythagorean's Theorem. you find a unit vector by taking the vector and dividing by its ownmagnitude. So: (in general a unit vector is found by using the formula:u = v/ ||v|| where u is theresulting unit vector and v is the vector youhave) r u = (i + j)/ ||i+j|| where i'musing bold instead of an arrow to represent a vector, and the || oneither side represents magnitude, which you find using PythagoreanTheorem (sometimes called the distance formula)...use thecoefficients of the i and j -->(12+12) = magnitude of the vector weused = (2) So the unit vector is 1/(2)i +1/(2)j = u Now we have a unit vector in the direction of NE, we need it tohave that displacement of 30.5 km, so multiply 30.5km to that unitvector 30.5km(1/(2))i +30.5km(1/(2))j is that unit vector. You can use thedistance formula to check that this has a magnitude of 30.5km. For the second part, follow the same process, but since he walksdue east we can just intuitively figure that the unit vector we'lluse is going to just be i. So take the 17 km and multiplyit by that vector (i)to get the desired displacement ofthat vector. so you have the vectors [(30.5km)/(2)]i +[(30.5km)/(2)]j and 17kmi so do somevector addition of the two vectors by adding like directions (youcan't add i to j). This will give you thedirection the guy walked. Then use the distance formula of thatfinal vector you found to find the displacement. Note: North taken as y direction (i), south is negative y(-i) and east is x direction (j) and west isnegative x (-j)

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