NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Three-dimensional coordinates

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A drone is flying above Brisbane at a position defined by the three-dimensional coordinates (2, -5, 6). Using the 3D distance formula, what is the exact distance from the drone to the origin (0, 0, 0) in metres?

Express your answer as an exact simplified radical expression.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The drone's position is given as (2, -5, 6) in a 3D coordinate system, with units in metres. The origin is (0, 0, 0). We are to find the straight-line distance from the drone to the origin using the 3D distance formula.
  2. The 3D distance formula is used to calculate the distance between two points in space. It extends the 2D distance formula by including the z-coordinate. The formula is: d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).
  3. Substitute the coordinates of the drone (2, -5, 6) and the origin (0, 0, 0) into the formula. Let (x1, y1, z1) = (0, 0, 0) and (x2, y2, z2) = (2, -5, 6).
  4. Calculate the differences: (2 - 0) = 2, (-5 - 0) = -5, (6 - 0) = 6.
  5. Square each difference: 2^2 = 4, (-5)^2 = 25, 6^2 = 36.
  6. Sum the squared differences: 4 + 25 + 36 = 65.
  7. Take the square root of the sum: sqrt(65).
  8. Check if 65 can be simplified under the square root. The prime factorisation of 65 is 5 × 13. Neither 5 nor 13 is a perfect square, so sqrt(65) cannot be simplified further.
  9. The distance is therefore exactly sqrt(65) metres.
  10. Verify the calculation: 2^2 = 4, (-5)^2 = 25, 6^2 = 36. 4 + 25 + 36 = 65. sqrt(65) is correct and in simplest radical form.
  11. The exact distance from the drone to the origin is sqrt(65) metres.
  12. The answer is expressed as an exact simplified radical, as required.

Reference answer: sqrt(65) m

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