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Chapter 12: Problem 7

Which point is farther from the origin, (3,-1,2) or (0,0,-4)\(?\)

Short Answer

Expert verified
Answer: Point B (0,0,-4) is farther from the origin than Point A (3,-1,2).

Step by step solution

01

Calculate the distance from the first point, (3,-1,2), to the origin

To calculate the distance from point (3,-1,2) to the origin (0,0,0), we will use the distance formula for three-dimensional coordinates, which is: Distance = \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\) For our point (3,-1,2) and the origin (0,0,0), the formula becomes: Distance_1 = \(\sqrt{(3 - 0)^2 + (-1 - 0)^2 + (2 - 0)^2}\)
02

Calculate the distance from the second point, (0,0,-4), to the origin

Similarly, to calculate the distance from point (0,0,-4) to the origin (0,0,0), we will use the distance formula for three-dimensional coordinates: Distance_2 = \(\sqrt{(0 - 0)^2 + (0 - 0)^2 + (-4 - 0)^2}\)
03

Compare the calculated distances

Now, we will compare the distances of both points to the origin: Distance_1 = \(\sqrt{(3 - 0)^2 + (-1 - 0)^2 + (2 - 0)^2} = \sqrt{9 + 1 + 4} = \sqrt{14}\) Distance_2 = \(\sqrt{(0 - 0)^2 + (0 - 0)^2 + (-4 - 0)^2} = \sqrt{0 + 0 + 16} = \sqrt{16} = 4\) As \(\sqrt{14}\) is approximately equal to 3.74 and 4 > 3.74, we can conclude that the second point (0,0,-4) is farther from the origin than the first point (3,-1,2).

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