If →u and →v are the vectors below, find the vector →w whose tail is at the point halfway from the tip of →v to the tip of →u and whose head is at the point halfway from the tip of →u to the tip of →u−→v. Assume all vectors are in standard position.

→u = [−5,−5,−5] →v = [4,−3,4] →w = [0,0,0]

To find the vector →w, we can use the vector addition formula and the midpoint formula.

Step 1: Find the midpoint of →v and →u:
The midpoint of →v and →u can be found by taking the average of their corresponding coordinates. Let's call this midpoint vector →m.

→m = [(v₁ + u₁)/2, (v₂ + u₂)/2, (v₃ + u₃)/2]

→m = [(4 + (-5))/2, (-3 + (-5))/2, (4 + (-5))/2]
= [(-1)/2, (-8)/2, (-1)/2]
= [-1/2, -4, -1/2]

Step 2: Find the midpoint of →u and →(u - v):
Similarly, we can find the midpoint of →u and →(u - v) and call it →n.

→n = [(u₁ + (u - v)₁) / 2, (u₂ + (u - v)₂) / 2, (u₃ + (u - v)₃) / 2]

→n = [(-5 + (-5 + 4))/2, (-5 + (-5 - 3))/2, (-5 + (-5 + 4))/2]
= [(-6)/2, (-13)/2, (-6)/2]
= [-3, -13/2, -3]

Step 3: Find the vector →w:
The vector →w will have its tail at →m and its head at →n.

→w = →n - →m

→w = [-3 - (-1/2), (-13/2) - (-4), -3 - (-1/2)]
= [-3 + 1/2, -13/2 + 4, -3 + 1/2]
= [-5/2, -5/2, -5/2]

Therefore, the vector →w is [-5/2, -5/2, -5/2].

To find the vector →w whose tail is at the midpoint of →v to the midpoint of →u and has its head at the midpoint of →u to the midpoint of →u−→v, we can follow these steps:

Step 1: Find the midpoint of →v
The midpoint of →v can be found by taking the average of the corresponding coordinates:
Midpoint of →v = [(x-coordinate of →v + x-coordinate of →u) / 2, (y-coordinate of →v + y-coordinate of →u) / 2, (z-coordinate of →v + z-coordinate of →u) / 2]

Midpoint of →v = [(4 + (-5)) / 2, (-3 + (-5)) / 2, (4 + (-5)) / 2]
Midpoint of →v = [-0.5, -4, -0.5]

Step 2: Find the midpoint of →u−→v
First, find →u−→v by subtracting the corresponding coordinates of →v from →u:
→u−→v = [(-5) - 4, (-5) - (-3), (-5) - 4]
→u−→v = [-9, -2, -9]

Then, find the midpoint of →u−→v by taking the average of the corresponding coordinates:
Midpoint of →u−→v = [(x-coordinate of →u−→v + x-coordinate of →u) / 2, (y-coordinate of →u−→v + y-coordinate of →u) / 2, (z-coordinate of →u−→v + z-coordinate of →u) / 2]

Midpoint of →u−→v = [(-9 + (-5)) / 2, (-2 + (-5)) / 2, (-9 + (-5)) / 2]
Midpoint of →u−→v = [-7, -3.5, -7]

Step 3: Find →w
Finally, →w can be found by taking the midpoint of the midpoint of →v to the midpoint of →u and the midpoint of →u to the midpoint of →u−→v:
→w = [(x-coordinate of midpoint of →v to midpoint of →u + x-coordinate of midpoint of →u to midpoint of →u−→v) / 2, (y-coordinate of midpoint of →v to midpoint of →u + y-coordinate of midpoint of →u to midpoint of →u−→v) / 2, (z-coordinate of midpoint of →v to midpoint of →u + z-coordinate of midpoint of →u to midpoint of →u−→v) / 2]

→w = [((-0.5) + (-7)) / 2, ((-4) + (-3.5)) / 2, ((-0.5) + (-7)) / 2]
→w = [-3.75, -3.75, -3.75]

Therefore, the vector →w with the given properties is →w = [-3.75, -3.75, -3.75].

the vector u-v is the vector from the tip of v to the tip of u.

You can see this, since v + u-v = u
so, the tip of u-v is the tip of u!

so, w = (u-v)/2