Fill in the blanks. The value of the expression | a b |^2+( a b )… - Vidyadip

Question

Fill in the blanks.
The value of the expression $|\vec{\text{a}}\times\vec{\text{b}}|^2+(\vec{\text{a}}\cdot\vec{\text{b}})^2$ is ________.

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Answer

The value of the expression $|\vec{\text{a}}\times\vec{\text{b}}|^2+(\vec{\text{a}}\cdot\vec{\text{b}})^2$ is $|\vec{\text{a}}|^2|\vec{\text{b}}|^2.$Solution:
$|\vec{\text{a}}\times\vec{\text{b}}|^2+(\vec{\text{a}}\cdot\vec{\text{b}})^2=|\vec{\text{a}}|^2|\vec{\text{b}}|^2\sin^2\theta+(\vec{\text{a}}\cdot\vec{\text{b}})^2$ $=|\vec{\text{a}}|^2|\vec{\text{b}}|^2(1-\cos^2\theta)+(\vec{\text{a}}\cdot\vec{\text{b}})^2$ $=|\vec{\text{a}}|^2|\vec{\text{b}}|^2-|\vec{\text{a}}|^2|\vec{\text{b}}|^2\cos^2\theta+(\vec{\text{a}}\cdot\vec{\text{b}})^2$ $=|\vec{\text{a}}|^2|\vec{\text{b}}|^2-(\vec{\text{a}}\cdot\vec{\text{b}})^2+(\vec{\text{a}}\cdot\vec{\text{b}})^2$ $|\vec{\text{a}}|^2|\vec{\text{b}}|^2+(\vec{\text{a}}\cdot\vec{\text{b}})^2=|\vec{\text{a}}|^2|\vec{\text{b}}|^2$

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