Question-6

Let \(a\) and \(b\) be two vectors. Compute the following:

• \(3(a+b)+[(a+b)-(a-b)]\)

• \(5(a+b)-[(a+b)-(a-b)]\)

• \(3(a+b)+[(a+b)+(a-b)]\)

• \(5(a+b)-[(a+b)+(a-b)]\)

---

First:

\[ \begin{aligned} 3(a+b)+[(a+b)-(a-b)] & =3a+3b+(a+b-a+b)\\ & =3a+3b+2b\\ & =3a+5b \end{aligned} \]

Second:

\[ \begin{aligned} 5(a+b)-[(a+b)-(a-b)] & =5a+5b-(a+b-a+b)\\ & =5a+5b-2b\\ & =5a+3b \end{aligned} \]

Third:

\[ \begin{aligned} 3(a+b)+[(a+b)+(a-b)] & =3a+3b+2a\\ & =5a+3b \end{aligned} \]

Fourth:

\[ \begin{aligned} 5(a+b)-[(a+b)+(a-b)] & =5a+5b-2a\\ & =3a+5b \end{aligned} \]