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 →v−→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.

Question

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 →v−→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 = (−2,3,3) →v = (1,3,1)→w = (-7/2,3,4) <---- my answer
I'm sorry for the constant posts about this question, please tell me if I got the correct answer. I've been wrapping my head over this problem. Thank you.

Answer 1

very good, that is what I got.

Btw, this is exactly the same type as your other question, only the numbers are different.

Answer 2

To find the vector →w, you need to calculate the coordinates of its tail and head points.

The tail point of →w is halfway from the tip of →v to the tip of →v - →u. To find this point, you can use the midpoint formula:

midpoint = (1/2) * (→v + (→v - →u))

Let's calculate it step by step:

Step 1: Find →v - →u
→v - →u = (1, 3, 1) - (-2, 3, 3)
= (1 + 2, 3 - 3, 1 - 3)
= (3, 0, -2)

Step 2: Add →v - →u to →v
→v + (→v - →u) = (1, 3, 1) + (3, 0, -2)
= (1 + 3, 3 + 0, 1 - 2)
= (4, 3, -1)

Step 3: Multiply the result by 1/2 to find the midpoint
(1/2) * (4, 3, -1) = (2, 3/2, -1/2)

The tail point of →w is (2, 3/2, -1/2).

Now, let's find the head point of →w, which is halfway from the tip of →u to the tip of →u + →v. Again, we can use the midpoint formula:

midpoint = (1/2) * (→u + (→u + →v))

Let's calculate it step by step:

Step 1: Find →u + →v
→u + →v = (-2, 3, 3) + (1, 3, 1)
= (-2 + 1, 3 + 3, 3 + 1)
= (-1, 6, 4)

Step 2: Add →u + →v to →u
→u + (→u + →v) = (-2, 3, 3) + (-1, 6, 4)
= (-2 - 1, 3 + 6, 3 + 4)
= (-3, 9, 7)

Step 3: Multiply the result by 1/2 to find the midpoint
(1/2) * (-3, 9, 7) = (-3/2, 9/2, 7/2)

The head point of →w is (-3/2, 9/2, 7/2).

Therefore, the vector →w has a tail at (2, 3/2, -1/2) and a head at (-3/2, 9/2, 7/2).

Answer 3

To find the vector →w, we need to calculate the points halfway between the tips of →v and →v-→u, as well as the points halfway between the tips of →u and →u+→v. Then we can subtract the coordinates of the first point from the coordinates of the second point to get the vector →w.

First, let's find the point halfway between the tips of →v and →v-→u:
→v: (1, 3, 1)
→v-→u: →v - →u = (1, 3, 1) - (-2, 3, 3) = (1, 3, 1) + (2, -3, -3) = (1+2, 3-3, 1-3) = (3, 0, -2)

To find the point halfway between the tips, we add the coordinates of →v and →v-→u together and divide each coordinate by 2:
(1/2)(1, 3, 1) + (1/2)(3, 0, -2) = (1/2)(1+3, 3+0, 1-2) = (1/2)(4, 3, -1) = (2, 3/2, -1/2)

Now let's find the point halfway between the tips of →u and →u+→v:
→u: (-2, 3, 3)
→u+→v: →u + →v = (-2, 3, 3) + (1, 3, 1) = (-2+1, 3+3, 3+1) = (-1, 6, 4)

To find the point halfway between the tips, we add the coordinates of →u and →u+→v together and divide each coordinate by 2:
(1/2)(-2, 3, 3) + (1/2)(-1, 6, 4) = (1/2)(-2-1, 3+6, 3+4) = (1/2)(-3, 9, 7) = (-3/2, 9/2, 7/2)

Finally, we subtract the coordinates of the first point from the coordinates of the second point to find →w:
→w = (-3/2, 9/2, 7/2) - (2, 3/2, -1/2) = (-3/2 - 2, 9/2 - 3/2, 7/2 - (-1/2)) = (-7/2, 6/2, 8/2) = (-7/2, 3, 4)

Thus, your answer of →w = (-7/2, 3, 4) is indeed correct.