Question

For three vectors $p , q$ and $r$, if $r =3 p +4 q$ and $2 r = p -3 q$, then

• A. $| r |<2| q |$ and $r , q$ have the same direction
• B. $| r |>2| q |$ and $r , q$ have opposite directions
• C. $| r |<2| q |$ and $r, q $ have opposite directions
• D. $| r |>2| q |$ and $r , q$ have the same direction

Answer 1

Answer: (B)

We have, $r=3 p+4 q\,\,\,...(i)$
and $2 r=p-3 q\,\,\,...(ii)$
On eliminating $p$, from Eqs. (i) and (ii), we get
$ r =3(2 r+3 q)+4 q$
$ r =6 r+9 q+4 q $
$\Rightarrow 5 r =-13 q $
$\Rightarrow r =-\frac{13}{5} q$
$\therefore r$ and $q$ have opposite directions.
Hence, $| r |>2| q |$