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Chapter 9: Q. 23 (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=3i+j,w=-2i-j

Short Answer

Expert verified

The values arev1=2.8i+1.4jandv2=0.2i-0.04j.

Step by step solution

01

Step 1. Given Information

The two vectors are v=3i+jandw=-2i-j.

We have to decomposevinto two vectorsv1andv2,wherev1is parallel towandv2is orthogonal tow.

02

Step 2. Finding v1

To find v1,we will use the formula v1=v路ww2w

Now,

v路w=3i+j-2i-jv路w=-6-1v路w=-7

And w=-22+-12w=4+1w=5w2=5

Now, let's put all the values in the formula

v1=-75-2i-jv1=-1.4-2i-jv1=2.8i+1.4j

03

Step 3. Finding v2

To find v2use the formula v2=v-v1

So,

v2=3i+j-2.8i+1.4jv2=3-2.8i+1-1.4jv2=0.2i-0.04j

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