Gandalf the grey started in the forest of mirkwood at a point with coordinates (-3, -3) and arrived in the iron hills at the point with coordinates (-2, 2). if he began walking in the direction of the vector v=3i+2j and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn.

v⃗ +w⃗ =z⃗  Now, you were given some component vector that relates to v⃗  so k1⟨3,2⟩=v⃗ .
And from ⟨3,2⟩ and 90 degrees you know w⃗ =k2⟨−2,3⟩ OR w⃗ =k2⟨2,−3⟩. And you can find z⃗  by looking a the math of the points... from (−3,−3) to (−2,2) is over 1 and up 5 so z⃗ =⟨1,5⟩.

Start and end are points, so (-3, -3) and (-2, 2).

v⃗ +w⃗ =z⃗ 
v⃗ =k1⟨3,2⟩
w⃗ =k2⟨−2,3⟩
z⃗ =⟨1,5⟩
k1⟨3,2⟩+k2⟨−2,3⟩=⟨1,5⟩

Authority is the accepted power of a person or an organization to lead another person or group. In a political sense, authority also stands for the power and dominion over an area or country held by the government. Authority legitimized by law or government is called 'competent authority'. A person who is generally respected by knowledge or status in a certain area is 'authoritative' in that area.

Authority regulates the relationships between people in society. If authority is absent, anarchy reigns. The difference between authority and power is that authority is legitimate. It has the right to exercise this power by, for example, imposing sanctions.