Question
Find the distance between (4, 2, 1) and (–2, 8, 4).
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15
9
7
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15
9
7
Answers
Answered by
GPT-4o mini
To find the distance between the points \((4, 2, 1)\) and \((-2, 8, 4)\), you can use the distance formula for points in 3D space:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
Here, let \((x_1, y_1, z_1) = (4, 2, 1)\) and \((x_2, y_2, z_2) = (-2, 8, 4)\).
Now, we can calculate each component:
1. \(x_2 - x_1 = -2 - 4 = -6\)
2. \(y_2 - y_1 = 8 - 2 = 6\)
3. \(z_2 - z_1 = 4 - 1 = 3\)
Now, plug these differences into the distance formula:
\[
d = \sqrt{(-6)^2 + (6)^2 + (3)^2}
\]
Calculating these squares:
- \((-6)^2 = 36\)
- \((6)^2 = 36\)
- \((3)^2 = 9\)
Now sum these values:
\[
d = \sqrt{36 + 36 + 9} = \sqrt{81} = 9
\]
Thus, the distance between the points \((4, 2, 1)\) and \((-2, 8, 4)\) is \(9\).
So, the answer is:
**9**
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
Here, let \((x_1, y_1, z_1) = (4, 2, 1)\) and \((x_2, y_2, z_2) = (-2, 8, 4)\).
Now, we can calculate each component:
1. \(x_2 - x_1 = -2 - 4 = -6\)
2. \(y_2 - y_1 = 8 - 2 = 6\)
3. \(z_2 - z_1 = 4 - 1 = 3\)
Now, plug these differences into the distance formula:
\[
d = \sqrt{(-6)^2 + (6)^2 + (3)^2}
\]
Calculating these squares:
- \((-6)^2 = 36\)
- \((6)^2 = 36\)
- \((3)^2 = 9\)
Now sum these values:
\[
d = \sqrt{36 + 36 + 9} = \sqrt{81} = 9
\]
Thus, the distance between the points \((4, 2, 1)\) and \((-2, 8, 4)\) is \(9\).
So, the answer is:
**9**
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