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Chapter 9: Q. 21 (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=i-jw=-i-2j

Short Answer

Expert verified

v1=-0.2i-0.4jv2=1.2i-0.6j

Step by step solution

01

Step 1. Concept Used  

The decomposition of vinto v1and v2, where v1is parallel to wand localid="1646827028342" v2is orthogonal to w, is

localid="1646827499665" v1=v路ww2wv2=v-v1

02

Step 2. Find the dot product and w

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

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

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

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

03

Step 3. Find v1 and v2

The vector v1which is parallel to wis given by the formulav1=v路ww2w.

On substituting the calculated values we get

v1=152(-i-2j)v1=15(-i-2j)v1=0.2(-i-2j)v1=-0.2i-0.4j

and the vector v2orthogonal to wis found as

v2=v-v1v2=i-j-(-0.2i-0.4j)v2=i-j+0.2i+0.4jv2=(1+0.2)i+(-1+0.4)jv2=1.2i-0.6j

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