$|\vec{u}-\vec{w}| \geq|\vec{u}|-\mid \vec{w} \|$ for all vectors $\vec{u}$ and $\vec{w}$

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Quizplus · Accepted Answer

This statement is a consequence of the triangle inequality, which in vector form states that the magnitude of the sum (or difference) of two vectors is less than or equal to the sum of the magnitudes of these vectors. Specifically, $|\vec{u}-\vec{w}|$ can be thought of as the length of one side of a triangle, while $|\vec{u}| + |\vec{w}|$ are the lengths of the other two sides. The inequality $|\vec{u}-\vec{w}| \geq |\vec{u}| - |\vec{w}|$ ensures that the difference in lengths does not exceed the actual distance between the endpoints of $\vec{u}$ and $\vec{w}$, adhering to the triangle inequality principle.

∣u⃗−w⃗∣≥∣u⃗∣−∣w⃗∥|\vec{u}-\vec{w}| \geq|\vec{u}|-\mid \vec{w} \|∣u−w∣≥∣u∣−∣w∥ For All Vectors u⃗\vec{u}u And w⃗\vec{w}w

$|\vec{u}-\vec{w}| \geq|\vec{u}|-\mid \vec{w} \|$ For All Vectors $\vec{u}$ And $\vec{w}$