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 = −4,−2,3 →v = −1,−1,−2 [→w = −11/2,−7/2,−3 <--- my answer]
The correct answer is:
→w = −11/2, −5/2, 11/2
I don't know what I am doing wrong..
Oh I see now, thank you so much!
Well, it looks like you're halfway there with your answer! But it seems there might be a little mix-up with the signs.
To find the vector →w, which lies halfway between the tip of →v and the tip of →v−→u, we can use the midpoint formula:
→w = 1/2 (→v + (→v−→u))
Let's simplify this:
→w = 1/2 (−1,−1,−2 + (−4,−2,3))
→w = 1/2 (−1 + −4, −1 + −2, −2 + 3)
→w = 1/2 (−5, −3, 1)
→w = (1/2)(−5, −3, 1)
→w = (−5/2, −3/2, 1/2)
So, the correct answer should be:
→w = −5/2, −3/2, 1/2
Oops, it seems the correct answer doesn't match what you said. My apologies, it looks like the clown bot made a mistake!
To find the vector →w, we need to first determine 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. Let's break down the steps to find →w:
Step 1: Calculate the halfway point between the tips of →v and →v-→u.
To find the midpoint between two points, we can use the average of their coordinates. In this case, the tips of →v and →v-→u are given by:
Tip of →v = (-1, -1, -2)
Tip of →v-→u = (-1, -1, -2) - (-4, -2, 3) = (3, 1, -5)
Using the average of the corresponding coordinates, we can find the midpoint:
Midpoint = [(x1 + x2) / 2, (y1 + y2) / 2, (z1 + z2) / 2]
= [(-1 + 3) / 2, (-1 + 1) / 2, (-2 - 5) / 2]
= [2 / 2, 0 / 2, -7 / 2]
= [1, 0, -7/2]
So, the halfway point between the tips of →v and →v-→u is (1, 0, -7/2).
Step 2: Calculate the halfway point between the tips of →u and →u+→v.
Using a similar approach, we find the tips of →u and →u+→v:
Tip of →u = (-4, -2, 3)
Tip of →u+→v = (-4, -2, 3) + (-1, -1, -2) = (-5, -3, 1)
Again, finding the average of the coordinates gives us the midpoint:
Midpoint = [(-4 - 5) / 2, (-2 - 3) / 2, (3 + 1) / 2]
= [-9 / 2, -5 / 2, 4 / 2]
= [-9 / 2, -5 / 2, 2]
So, the halfway point between the tips of →u and →u+→v is (-9/2, -5/2, 2).
Step 3: Find the vector →w connecting the two halfway points.
To find →w, we subtract the coordinates of the tail from the coordinates of the head:
→w = [head coordinates] - [tail coordinates]
= (-9/2, -5/2, 2) - (1, 0, -7/2)
= (-9/2 - 1, -5/2 - 0, 2 - (-7/2))
= (-11/2, -5/2, 11/2)
Therefore, the correct answer for →w is (-11/2, -5/2, 11/2), which matches the given solution.
Tip of V = -1 , -1 , -2
Tip V-U = +3 , +1 , -5
halfway V to V-U is average
= (3-1)/2 , 0/2 , (-2-5)/2 = +1 , 0 , -7/2
Tip of U = -4 , -2 , +3
Tip V+U= -5 , -3 , +1
halfsum = -9/2 , -5/2 , 2
now finish
-9/2 - 1 = -11/2
-5/2 - 0 = -5/2
2 - - 7/2 = 11/2