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⟩

Ai just took the test