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WZ
Numerade Educator

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Problem 62 Hard Difficulty

The Triangle Inequality for vectors is
$$ \mid a + b \mid \le \mid a \mid + \mid b \mid $$
(a) Give a geometric interpretation of the Triangle Inequality.
(b) Use the Cauchy-Schwarz Inequality from Exercise 61 to prove the Triangle Inequality. [Hint: Use the fact that $ {\mid a + b \mid}^2 = (a + b) \cdot (a + b) $ and use Property 3 of the dot product.]

Answer

a) Therefore the triangle inequality says that the length of oe side of a triangle is less than the sum of the lengths of the other two sides.
b) $|\mathbf{a}+\mathbf{b}|^{2}=(\mathbf{a}+\mathbf{b}) \cdot(\mathbf{a}+\mathbf{b})$
$|\mathbf{a}+\mathbf{b}|^{2} \leq(|\mathbf{a}|+|\mathbf{b}|)^{2}$
$|\mathbf{a}+\mathbf{b}| \leq(|\mathbf{a}|+|\mathbf{b}|) \quad(\mathrm{R} . \mathrm{T} . \mathrm{P})$

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Video Transcript

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