Let , , and , where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then is equal to __________. [2024]

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Solution

Q.
Let $\vec{O A} = \vec{a}$ , $\vec{O B} = 12 \vec{a} + 4 \vec{b}$ and $\vec{O C} = b$ , where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then $\frac{area of the quadrilateral O A B C}{area of S}$ is equal to __________. [2024]

1 8

2 10

3 7

4 6

Ans.
(1)

We have,

Area of parallelogram, $S =$ $| \vec{O A} \times \vec{O C} |$ $=$ $| \vec{a} \times \vec{b} |$ ... (i)

Area of quad. OABC = Area of $△$ OAB + Area of $△$ OBC

$= \frac{1}{2}$ $| \vec{O A} \times \vec{O B} |$ $+ \frac{1}{2}$ $| \vec{O B} \times \vec{O C} |$

$= \frac{1}{2}$ $| \vec{a} \times (12 \vec{a} + 4 \vec{b}) |$ $+ \frac{1}{2}$ $| (12 \vec{a} + 4 \vec{b}) \times \vec{b} |$

$= \frac{1}{2} \times (4 | \vec{a} \times \vec{b} | + 12 | \vec{a} \times \vec{b} |) = 8$ $| \vec{a} \times \vec{b} |$

$∴ \frac{Area of quad . O A B C}{Area of parallelogram} = \frac{8 | \vec{a} \times \vec{b} |}{| \vec{a} \times \vec{b} |} = 8$ .

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