Chapter 11, Problem 1PS

a.

To determine

To draw: u and v.

Answer to Problem 1PS

  

Explanation of Solution

Given information: Let $u=i+j,v=j+k,w=au+bv.$

Calculation:

  $u=i+j,v=j+k,w=au+bv.$

Sketch of u and v is given below.

  

b.

To determine

To show: When w=0 , a and b must be zero.

Answer to Problem 1PS

  $a=b=0$ .

Hence proved.

Explanation of Solution

Given information: Let $u=i+j,v=j+k,w=au+bv.$

Calculation:

  $u=i+j,v=j+k,w=au+bv.$

  $\begin{array}{l}w=au+bv=0.\\ a(i+j)+b(j+k)=0.\\ ai+aj+bj+bk=0.\\ ai+(a+b)j+bk=0i+0j+0k.\\ a=0.\\ a+b=0\\ b=0.\\ a=b=0\end{array}$

Hence proved.

c.

To determine

To find: $a\text{and}b\text{such}\text{that}w=i+2j+k.$

Answer to Problem 1PS

  $a=b=1.$

Explanation of Solution

Given information: Let $u=i+j,v=j+k,w=au+bv.$

Calculation:

  $\begin{array}{l}u=i+j,v=j+k,w=au+bv.\\ w=au+bv=i+2j+k.\\ a(i+j)+b(j+k)=i+2j+k.\\ ai+aj+bj+bk=i+2j+k.\\ ai+(a+b)j+bk=i+2j+k.\\ So,\\ a=1.\\ a+b=2.\\ b=1.\\ a=b=1.\end{array}$

d.

To determine

To show: No choice of $a\text{and}byieldsw=i+2j+3k.$

Answer to Problem 1PS

So, $2\ne 4$ that no choice of $a\text{and}byieldsw=i+2j+3k.$

Hence Proved.

Explanation of Solution

Given information: Let $u=i+j,v=j+k,w=au+bv.$

Calculation:

  $\begin{array}{l}u=i+j,v=j+k,w=au+bv.\\ w=au+bv=i+2j+3k.\\ a(i+j)+b(j+k)=i+2j+3k.\\ ai+aj+bj+bk=i+2j+3k.\\ ai+(a+b)j+bk=i+2j+3k.\\ So,\\ a=1.\\ a+b=2.\\ b=3.\\ a+b=1+3=4\\ a+b=2\\ 2=4.\\ \end{array}$

So, $2\ne 4$ that no choice of $a\text{and}byieldsw=i+2j+3k.$

Hence Proved.