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# 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.]

## 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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##### Top Calculus 3 Educators ##### Lily A.

Johns Hopkins University  ##### Heather Z.

Oregon State University Lectures

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