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Chapter 10: Q .32. (page 812)

In Exercises 32鈥�36,

(a) computeu.v

(b) find the angle between u and v, and

(c) findprojuv

Short Answer

Expert verified

Part a)u.v=37

Part b) angle is 胃=cos-1371378

Part c)The value ofprojuvis3726鉄�1,5鉄�.

Step by step solution

01

Step 1:Given information Part a)

u and v are vectors

02

Part a) Simplification

Consider the vectoru=鉄�1,5鉄�,v=鉄�2,7鉄�.

Ifuandvare the vector such thatu=u1,v1andv=u2,v2, then dot product is given by

u.v=u1v1+u2v2.

u路v=1(2)+5(7)

=37

03

Step 3:Part b) Given information

u,vare given vectors

04

Step 4:Part b)Simplification

u.v=37

Also,u=鉄�1,5鉄�

Therefore

鈥�u鈥�=12+52

=26

Also,v=鉄�2,7鉄�

鈥�v鈥�=22+72

=53

now,

cos胃=372653

鈬�胃=cos-1371378

Hence, the angle between the vectors is cos-1371378

05

Part c):Given information

the vectoru=鉄�1,5鉄�,v=鉄�2,7鉄�

06

Step 6:Part c) Simplification

鈬�u.v=37calculated

and

鈥�u鈥�=26

Letube any non- zero vector, then the vector projection ofvontouis given by

projuv=u路v鈥�u鈥�2u路Now, substitute the values inprojuv=u路v鈥�u鈥�2u

projuv=37(26)2鉄�1,5鉄�

=3726鉄�1,5鉄�

Hence, the value ofprojuvis3726鉄�1,5鉄�.

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