Question

The area of the parallelogram with $\vec{a}$ and $\vec{b}$ as adjacent sides is $20\, sq. \,units$. Then the area of the parallelogram having $7\vec{a} + 5\vec{b}$ and $8\vec{a} + 11\vec{b}$ as adjacent sides is

• A. $2960\, sq.\, units$
• B. $740\, sq.\, units$
• C. $1340\, sq.\, units$
• D. $3400\, sq.\, units$

Answer 1

Answer: (B)

Area of parallelogram having adjacent sides $\vec{a}$ and $\vec{b}$ = 20 sq. units
$\Rightarrow \:\: | \vec{a} \times \vec{b}| = 20$
Now, area of parallelogram
$ = |(7\vec{a} + 5 \vec{b}) \times ( 8 \vec{a} + 11\vec{b})|$
$= \left|56\left(\vec{a} \times \vec{a}\right)+77\left(a \times \vec{b}\right)+40\left(\vec{b}\times \vec{a}\right)+55\left(\vec{b}\times \vec{b}\right)\right| $
$= \left|77\left(\vec{a} \times \vec{b}\right)-40\left(\vec{a}\times \vec{b}\right)\right| $
$= \left|77 -40\right|\left|a\times b\right|= \left(37\times 20\right) = 740$ sq. units