-a-b-5 and the angle between a and b is -4- The area of the triangle constructed on the vectors a-2b and 3a-2b is-

Finding the area of the triangle:Given, |a|=|b|=5 and angle between a&bis π/4. Also the sides a 2b and 3a+2b.By using formula for area of triangle is 1/2×|AB×BC ...

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Standard XII

Physics

Cross Product

|a|=|b|=5 and...

Question

$| \vec{a} | = | \vec{b} | = 5$ and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{π}{4}$ . The area of the triangle constructed on the vectors $\vec{a} - 2 \vec{b}$ and $3 \vec{a} + 2 \vec{b}$ is?

$50$

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$50 \sqrt{2}$

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$\frac{50}{\sqrt{2}}$

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$100$

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Solution

The correct option is B
$50 \sqrt{2}$

Finding the area of the triangle:
Given, $| \vec{a} | = | \vec{b} | = 5$ and angle between $\vec{a} & \vec{b}$ is $\frac{π}{4}$ . Also the sides $\vec{a} - 2 \vec{b}$ and $3 \vec{a} + 2 \vec{b}$ .
By using formula for area of triangle is $\frac{1}{2} \times | \vec{AB} \times \vec{BC} |$
$= \frac{1}{2} \times |(\vec{a} - 2 \vec{b}) \times (3 \vec{a} + 2 \vec{b})| = \frac{1}{2} \times |3 (\vec{a} \times \vec{a}) + 2 (\vec{a} \times \vec{b}) - 6 (\vec{b} \times \vec{a}) - 4 (\vec{b} \times \vec{b})| = \frac{1}{2} \times |2 \times 0 + 2 (\vec{a} \times \vec{b}) - 6 (\vec{b} \times \vec{a}) - 4 \times 0| (∵ |\vec{a} \times \vec{a}| = 0 = |\vec{b} \times \vec{b}|) = \frac{1}{2} \times |(2 (\vec{a} \times \vec{b}) + 6 (\vec{a} \times \vec{b}))| (∵ \vec{a} \times \vec{b} = - (\vec{b} \times \vec{a})) = \frac{1}{2} \times |(8 (\vec{a} \times \vec{b}))| = 4 |\vec{a}| |\vec{b}| \sin (θ) (∵ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| s i n (θ)) = 4 \times 5 \times 5 \times \sin (\frac{π}{4}) = 4 \times 5 \times 5 \times \frac{1}{\sqrt{2}} = \frac{100}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = 50 \sqrt{2}$
Therefore, correct answer is option $(B) .$

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|a→|=|b→|=5 and the angle between a→ and b→ is π4. The area of the triangle constructed on the vectors a→-2b→ and 3a→+2b→ is?

$| \vec{a} | = | \vec{b} | = 5$ and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{π}{4}$ . The area of the triangle constructed on the vectors $\vec{a} - 2 \vec{b}$ and $3 \vec{a} + 2 \vec{b}$ is?