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Chapter 9: Q. 22 (page 612)

In Problems 19鈥�24, decompose v into two vectors v1and v2, where v1 is parallel to w and v2 is orthogonal to w.

v=2i-jw=i-2j

Short Answer

Expert verified

v1=0.8i-1.6jv2=1.2i+0.6j

Step by step solution

01

Step 1. Concept Used 

The decomposition of vinto v1and v2, where v1is parallel to wand v2is orthogonal to w, is

role="math" localid="1646827695125" v1=v路ww2wv2=v-v1

02

Step 2. Find the dot product and w

For the vectors v=2i-jand w=i-2jthe dot product is given as

v路w=(2i-j)路(i-2j)v路w=2路1+(-1)路(-2)v路w=2+2v路w=4

The magnitude of vector w=i-2jis given as

w=12+(-2)2w=1+4w=5

03

Step 3. Find v1 and v2

The vector v1which is parallel to wis given by the formula v1=v路ww2w.

On substituting the calculated values we get

role="math" localid="1646827745681" v1=452(i-2j)v1=45(i-2j)v1=0.8(i-2j)v1=0.8i-1.6j

and the vector v2orthogonal to wis found as

v2=v-v1v2=2i-j-(0.8i-1.6j)v2=2i-j-0.8i+1.6jv2=(2-0.8)i+(-1+1.6)jv2=1.2i+0.6j

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