Question

Give an algebraic proof for the triangle inequality

$\mathbf{\left|}\left|\stackrel{\rightharpoonup }{v}+\stackrel{\rightharpoonup }{w}\right|\mathbf{\right|}\mathbf{\le }\mathbf{\left|}\left|\stackrel{\rightharpoonup }{v}\right|\mathbf{\right|}\mathbf{+}\mathbf{\left|}\left|\stackrel{\rightharpoonup }{w}\right|\mathbf{\right|}$

Draw a sketch. Hint: Expand${\left|\left|\stackrel{\rightharpoonup }{v}+\stackrel{\rightharpoonup }{w}\right|\right|}^{\mathbf{2}}\mathbf{=}\left(\stackrel{\rightharpoonup }{v}+\stackrel{\rightharpoonup }{w}\right)\mathbf{·}\left(\stackrel{\rightharpoonup }{v}+\stackrel{\rightharpoonup }{w}\right)$

Then use the Cauchy–Schwarz inequality.

Short answer

The triangle inequality is confirmed.

Step-by-step solution

Define Cauchy–Schwarz inequality.

If $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{x}}$and$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{y}}$are vectors in Rⁿthen

$\left|\stackrel{\rightharpoonup }{x}·\stackrel{\rightharpoonup }{y}\right|\mathbf{\le }\left|\left|\stackrel{\rightharpoonup }{x}\right|\right|\mathbf{·}\left|\left|\stackrel{\rightharpoonup }{y}\right|\right|$

This statement is an equality if (and only if)$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{x}}$and$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{y}}$are parallel.

Determine the triangle inequality and draw the sketch

Use the hint:

$$\begin{gathered} {\left||\stackrel{\rightharpoonup }{\mathrm{v}}+\stackrel{\rightharpoonup }{\mathrm{w}}|\right|}^2=|\left|\stackrel{\rightharpoonup }{\mathrm{v}}\right|{|}^2+{\left|\left|\stackrel{\rightharpoonup }{\mathrm{w}}\right|\right|}^2+2\left(\stackrel{\rightharpoonup }{\mathrm{v}}·\stackrel{\rightharpoonup }{\mathrm{w}}\right) \\ {\left||\stackrel{\rightharpoonup }{\mathrm{v}}+\stackrel{\rightharpoonup }{\mathrm{w}}|\right|}^2\le |\left|\stackrel{\rightharpoonup }{\mathrm{v}}\right|{|}^2+{\left|\left|\stackrel{\rightharpoonup }{\mathrm{w}}\right|\right|}^2+2\left|\left|\stackrel{\rightharpoonup }{\mathrm{v}}\right|\right|\left|\left|\stackrel{\rightharpoonup }{\mathrm{w}}\right|\right| \\ {\left||\stackrel{\rightharpoonup }{\mathrm{v}}+\stackrel{\rightharpoonup }{\mathrm{w}}|\right|}^2={\left(\left|\left|\stackrel{\rightharpoonup }{\mathrm{v}}\right|\right|+\left|\left|\stackrel{\rightharpoonup }{\mathrm{w}}\right|\right|\right)}^2 \\ {\left||\stackrel{\rightharpoonup }{\mathrm{v}}+\stackrel{\rightharpoonup }{\mathrm{w}}|\right|}^2\le {\left(\left|\left|\stackrel{\rightharpoonup }{\mathrm{v}}\right|\right|+\left|\left|\stackrel{\rightharpoonup }{\mathrm{w}}\right|\right|\right)}^2 \end{gathered}$$

The sketch for the inequality is shown below:

Thus, by taking square root of both sides, triangle inequality is confirmed.

Hence, the triangle inequality is confirmed.