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If the area of the parallelogram with $\mathbf{a}$ and $\mathbf{b}$ as two adjacent sides is 15 sq units, then the area of the parallelogram having $3 \mathbf{a}+2 \mathbf{b}$ and $\mathbf{a}+3 \mathbf{b}$ as two adjacent sides in sq units is
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105
Area of parallelogram having $\mathbf{a}$ and $\mathbf{b}$ as its adjacent sides is 15 sq units.
$\Rightarrow \quad|\mathbf{a} \times \mathbf{b}|=15$
Area of parallelogram having $(3 \mathbf{a}+2 \mathbf{b})$ and
$(\mathbf{a}+3 \mathbf{b})$ as two adjacent side
$=|(3 \mathbf{a}+2 \mathbf{b}) \times(\mathbf{a}+3 \mathbf{b})|$
$=|3 \mathbf{a} \times \mathbf{a}+2 \mathbf{b} \times \mathbf{a}+9 \mathbf{a} \times \mathbf{b}+6 \mathbf{b} \times \mathbf{b}|$
$=|2 \mathbf{b} \times \mathbf{a}+9 \mathbf{a} \times \mathbf{b}|$
$[\therefore \mathbf{a} \times \mathbf{a}=\mathbf{b} \times \mathbf{b}=0]$
$=7|\mathbf{a} \times \mathbf{b}|$
$[\therefore \mathbf{b} \times \mathbf{a}=-\mathbf{a} \times \mathbf{b}]$
$=7 \times 15$
$=105$
$\Rightarrow \quad|\mathbf{a} \times \mathbf{b}|=15$
Area of parallelogram having $(3 \mathbf{a}+2 \mathbf{b})$ and
$(\mathbf{a}+3 \mathbf{b})$ as two adjacent side
$=|(3 \mathbf{a}+2 \mathbf{b}) \times(\mathbf{a}+3 \mathbf{b})|$
$=|3 \mathbf{a} \times \mathbf{a}+2 \mathbf{b} \times \mathbf{a}+9 \mathbf{a} \times \mathbf{b}+6 \mathbf{b} \times \mathbf{b}|$
$=|2 \mathbf{b} \times \mathbf{a}+9 \mathbf{a} \times \mathbf{b}|$
$[\therefore \mathbf{a} \times \mathbf{a}=\mathbf{b} \times \mathbf{b}=0]$
$=7|\mathbf{a} \times \mathbf{b}|$
$[\therefore \mathbf{b} \times \mathbf{a}=-\mathbf{a} \times \mathbf{b}]$
$=7 \times 15$
$=105$
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