Question

Let $u=5a+6b+7c,v=7a-8b+9c$ and $w=3a+20b+5c$, where $a,b$ and $c$ are non-zero vectors. If $u=lv+mw$, then the values of $l$ and $m$ respectively are

• A. $\frac{1}{2},\frac{1}{2}$
• B. $\frac{1}{2},-\frac{1}{2}$
• C. $-\frac{1}{2},\frac{1}{2}$
• D. $\frac{1}{3},\frac{1}{3}$
• E. $-\frac{1}{2},\frac{1}{5}$

Answer 1

Answer: (A)

$\frac{1}{2},\frac{1}{2}$

Given, $u=5a+6b+7c$
$v=7a-8b+9c$
and $w=3a+20b+5c$
Now, $u=lv+mw$
$\Rightarrow (5a+6b+7c)=l(7a-8b+9c)+m(3a+20b+5c)$
$\Rightarrow (5a+6b+7c)=(7l+3m)a+(20m-8l)b+(5m+9l)c$
On comparing, we get
$7l+3m=5$ .... (i)
$20m-8l=6$ .... (ii)
and $5m+9l=7$ .... (iii)
On multiply by $4$ in Eq. (iii) and then subtracting from Eq. (ii), we get
$-8l-36l=6-28$
$\Rightarrow -44l=-22$
Therefore, $l=\frac{1}{2}$ then from Eq. (ii), $m=\frac{1}{2}$.