alleryThe distance between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ in three-dimensional space is:
A$\sqrt{(x_2 + x_1)^2 + (y_2 + y_1)^2 + (z_2 + z_1)^2}$
B$(x_2 - x_1)(y_2 - y_1)(z_2 - z_1)$
C$|x_2 - x_1| + |y_2 - y_1| + |z_2 - z_1|$
D$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
Answer & Solution
Correct answer: D. $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
3D distance formula extends Pythagoras into three dimensions: the distance is the square root of the sum of squared coordinate differences.
Derivation: project the segment $PQ$ onto each axis to get the leg lengths $|x_2 - x_1|$, etc. Apply Pythagoras twice.
Option B is the **Manhattan** (taxicab) distance, useful in some lattice problems but not the Euclidean answer.
Keep practising Unit 8A: Areas Between Curves
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