Exercise · ZW-METRIC-SPACE-008 · ★★☆ Intermediate
In \(\mathbb{R}^3\) with the Euclidean distance, what is the distance between \((3,4,8)\) and \((1,1,2)\)?
Choose one option.
Show answer
- B. 7
Explanation
The differences are \(2,3,6\) and \(\sqrt{4+9+36}=\sqrt{49}=7\). The value 49 forgets the root.
Review the concept: Metric space
A metric space is a set \(X\) together with a distance \(d\) between its points: a nonnegative number for each pair, symmetric, zero only between a point and itself, and satisfying the triangle inequality.
\[\begin{aligned}d(x,y)&=0\iff x=y,\qquad d(x,y)=d(y,x),\\ d(x,z)&\le d(x,y)+d(y,z)\end{aligned}\]
More practice problems: metric space
- ZW-METRIC-SPACE-001 In ℝ^2 with the Euclidean distance, what is the distance between (1,2) and (4,6)?
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- ZW-METRIC-SPACE-003 With the max distance d_∞(P,Q)=max\|p_1-q_1|,|p_2-q_2|\, what is d_∞((1,2),(4,6))?
- ZW-METRIC-SPACE-004 Let X carry the discrete metric (d(a,b)=1 if ane b and d(a,a)=0). For three distinct points a,b,c,…
- ZW-METRIC-SPACE-005 In a metric space, if d(x,y)=0, what follows?
- ZW-METRIC-SPACE-006 Why is d(x,y)=|x^2-y^2| not a metric on ℝ?
- ZW-METRIC-SPACE-007 Let d(x,y)=(x-y)^2 on ℝ. For the points 0,1,2, what is d(0,2)-big(d(0,1)+d(1,2)big)?
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