Let | a | = 2 , | b | = 3 and the angle between the vectors a and b be 4 . Then | ( a + 2 b ) × ( 2 a - 3 b ) | 2 is equal to [2023]

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Solution

Q.
Let $| \vec{a} |$ $= 2,$ $| \vec{b} |$ $= 3$ and the angle between the vectors $\vec{a}$ and $\vec{b}$ be $\frac{π}{4}$ . Then $| (\vec{a} + 2 \vec{b}) \times (2 \vec{a} - 3 \vec{b}) |^{2}$ is equal to [2023]

1 882

2 841

3 441

4 482

Ans.
(1)

We have,

$| (\vec{a} + 2 \vec{b}) \times (2 \vec{a} - 3 \vec{b}) |^{2}$ $= | - 3 (\vec{a} \times \vec{b}) + 4 (\vec{b} \times \vec{a}) |^{2}$

$= | - 7 (\vec{a} \times \vec{b}) |^{2}$ $= 49 | \vec{a} |^{2} | \vec{b} |^{2} \sin^{2} θ$ $= 49 \times 4 \times 9 \times \frac{1}{2} = 882$

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