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The video uses the analogy of commuting from work to home. Going straight home represents the direct distance $d(x,z)$.

Conditions: $x$ is work, $z$ is home, and $y$ is the shop.; The path is a journey in a metric space.

The video uses the analogy of commuting from work to home. Going straight home represents the direct distance $d(x,z)$.

Conditions: $x$ is work, $z$ is home, and $y$ is the shop.; The path is a journey in a metric space.

The video explains the triangle inequality using a detour analogy. It compares going straight from point $x$ to point $z$ with taking a detour through an intermediate point $y$.

Conditions: $x, y, z$ are points in the metric space.; $y$ is an arbitrary intermediate point.

The video explains the triangle inequality using a detour analogy. It compares going straight from point $x$ to point $z$ with taking a detour through an intermediate point $y$.

Conditions: $x, y, z$ are points in the metric space.; $y$ is an arbitrary intermediate point.