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Chapter 9: Q. 41 (page 613)

Let vand wdenote two nonzero vectors. Show that the vectorswv+vwandwv-vware orthogonal.

Short Answer

Expert verified

The given vectors are orthogonal.

Step by step solution

01

Step 1. Given Information 

There are two nonzero vectors vand w.

We have to show that the vectorswv+vwandwv-vware orthogonal.

02

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 wv+vwandwv-vw.

So,

role="math" localid="1646821115195" (||w||v+||v||w)路(||w||v-||v||w)=||w||2v路v-||w||||v||v路w+||v||||w||w路v-||v||2w路w=||w||2v路v-||v||2w路w=||w||2||v||2-||v||2||w||2[a路a=a2]=0

Therefore the given vectors are orthogonal.

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