Vectors---help!?

Question

I have two questions regarding vectors. Can someone please help.

1)A person desires to reach a point that is 6.10 km from her present location and in a direction that is 35.0° north of east. However, she must travel along streets that are oriented either north-south or east-west. What is the minimum distance she could travel to reach her destination?

2)A car is driven east for a distance of 46 km, then north for 34 km, and then in a direction 26° east of north for 29 km. Draw the vector diagram and determine the total displacement of the car from its starting point.
find Magnitude

3) Vectors:
a = (5.0 m) i + (3.0 m) j
b = (-10 m) i + (3.0 m) j
What is the direction of a + b?


Find Direction ° (counterclockwise from east)

Answer 1

Such problems can be solved systematically with a coordinate system. Let us have x-axis pointing to the east and y-axis pointing to the north. As you know, the unit vector along the x-axis is i and along the y-axis is j.
(1) from simple trig, the target point is at (in unit km):
6.10*cos35.0° i + 6.10*sin35.0° j
The problem says that she must walk along i and j. Hence the minimum distance is:
6.10*(cos35.0° + sin35.0°) (km)
(2) Let unit to be km. Drive east: 46 i,
Drive north: 34 j,
Drive in a direction 26° east of north: 29*sin26° i + 29*cos26° j, and
Overall: (46 + 29*sin26°)i + (34 + 29*cos26°)j = Ai + Bj
The total displacement of the car from its starting point is:
sqrt(A^2 + B^2).
(3) a+b = (-5.0m)i + (6.0m)j
Let the direction be (theta) north of east. We know:
tan(theta) = 6.0/(-5.0)
thus: theta = tan^(-1)[6.0/(-5.0)]
Please go ahead to get numerical data.