1. Show geometrically that (vectors u & v) ||u|-|v||</=|u+v|. Under what conditions does the quality hold?

2. find the altitude AD in the triangle ABC, for the points A (2,3/2,-4), B(3,-4,2) and C (1,3,-7).





Please show all logical work and reasoning.

1. Show geometrically that (vectors u %26amp; v) ||u|-|v||%26lt;/=|u+v|. Under what conditions does the quality hold?
1. Clearly,





|u| = |(u - v) + v| ≤ |u - v| + |v| ⇒ |u| - |v| ≤ |u - v|





by the triangle inequality. Similarly,





|v| = |(v - u) + u| ≤ |v - u| + |u| = |u - v| + |u| ⇒ |v| - |u| ≤ |u - v|.





Hence,





- |u - v| ≤ |u| - |v| ≤ |u - v|,





which is equivalent to





||u| - |v|| ≤ |u - v|.





Considering -v instead of v yields the equivalent answer





||u| - |v|| ≤ |u + v|.

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