Question

Let vand wdenote two nonzero vectors. Show that the vectors$\left|\left|\mathit{w}\right|\right|\mathit{v}+\left|\left|\mathit{v}\right|\right|\mathit{w}and\left|\left|\mathit{w}\right|\right|\mathit{v}-\left|\left|\mathit{v}\right|\right|\mathit{w}$are orthogonal.

Short answer

The given vectors are orthogonal.

Step-by-step solution

Step 1. Given Information 

There are two nonzero vectors vand w.

We have to show that the vectors$\left|\left|\mathit{w}\right|\right|\mathit{v}\mathbf{}+\left|\left|\mathit{v}\right|\right|\mathit{w}\mathbf{}and\left|\left|\mathit{w}\right|\right|\mathit{v}\mathbf{}-\left|\left|\mathit{v}\right|\right|\mathit{w}$are orthogonal.

Step 2. Showing the vectors are orthogonal

As we know if any two vectors are orthogonal then their dot product is equal to zero.

Let's find the dot product of $\left|\left|\mathit{w}\right|\right|\mathit{v}\mathbf{}+\left|\left|\mathit{v}\right|\right|\mathit{w}\mathbf{}and\left|\left|\mathit{w}\right|\right|\mathit{v}\mathbf{}-\left|\left|\mathit{v}\right|\right|\mathit{w}\mathbf{.}$

So,

role="math" localid="1646821115195" $$\begin{gathered} \mathbf{(}\mathbf{\left|}\left|w\right|\mathbf{\right|}\mathbf{v}\mathbf{}\mathbf{+}\mathbf{}\mathbf{\left|}\left|v\right|\mathbf{\right|}\mathbf{w}\mathbf{)}·\mathbf{(}\mathbf{\left|}\left|w\right|\mathbf{\right|}\mathbf{v}\mathbf{}\mathbf{-}\mathbf{\left|}\left|v\right|\mathbf{\right|}\mathbf{w}\mathbf{)} \\ ={\mathbf{\left|}\mathbf{\left|}\mathbf{w}\mathbf{\right|}\mathbf{\right|}}^2\mathbf{v}\mathbf{·}\mathbf{v}\mathbf{}-\mathbf{\left|}\mathbf{\left|}\mathbf{w}\mathbf{\right|}\mathbf{\right|}\mathbf{\left|}\mathbf{\left|}\mathbf{v}\mathbf{\right|}\mathbf{\right|}\mathbf{v}\mathbf{·}\mathbf{w}\mathbf{}\mathbf{+}\mathbf{\left|}\mathbf{\left|}\mathbf{v}\mathbf{\right|}\mathbf{\right|}\mathbf{\left|}\mathbf{\left|}\mathbf{w}\mathbf{\right|}\mathbf{\right|}\mathbf{w}\mathbf{·}\mathbf{v}\mathbf{}-\mathbf{|}\mathbf{\left|}\mathbf{v}\mathbf{\right|}{\mathbf{|}}^{\mathbf{2}}\mathbf{w}\mathbf{·}\mathbf{w} \\ ={\mathbf{\left|}\mathbf{\left|}\mathbf{w}\mathbf{\right|}\mathbf{\right|}}^2\mathbf{v}\mathbf{·}\mathbf{v}-\mathbf{|}\mathbf{\left|}\mathbf{v}\mathbf{\right|}{\mathbf{|}}^{\mathbf{2}}\mathbf{w}\mathbf{·}\mathbf{w} \\ ={\mathbf{\left|}\mathbf{\left|}\mathbf{w}\mathbf{\right|}\mathbf{\right|}}^2\mathbf{|}\mathbf{\left|}\mathbf{v}\mathbf{\right|}{\mathbf{|}}^{\mathbf{2}}-\mathbf{|}\mathbf{\left|}\mathbf{v}\mathbf{\right|}{\mathbf{|}}^{\mathbf{2}}{\mathbf{\left|}\mathbf{\left|}\mathbf{w}\mathbf{\right|}\mathbf{\right|}}^{\mathbf{2}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left[a·a={\left|\left|a\right|\right|}^2\right] \\ =0 \end{gathered}$$

Therefore the given vectors are orthogonal.