Question

The area of the parallelogram with $\vec{a}$ and $\vec{b}$ as adjacent sides is $20sq.units$. 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. $2960sq.units$
• B. $740sq.units$
• C. $1340sq.units$
• D. $3400sq.units$

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