Question

Assertion : If $(\vec{a} \times \vec{b})^{2}+(\vec{a} \cdot \vec{b})^{2}=144$ and $|\vec{a}|=4$, then $|\vec{b}|=9$
Reason : If $\vec{ a }$ and $\vec{ b }$ are any two vectors, then $(\vec{ a } \times \vec{ b })^{2}$ is equal to $(\vec{a})^{2}(\vec{b})^{2}-(\vec{a} \cdot \vec{b})^{2}$

• A. Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
• B. Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
• C. Assertion is correct, Reason is incorrect
• D. Assertion is incorrect, Reason is correct.

Answer 1

Answer: (D)

$(\vec{a} \times \vec{b})^{2}+(\vec{a} \cdot \vec{b})^{2}=144|\vec{a}|=4$
We know that $(\vec{a} \times \vec{b})^{2}+(\vec{a} \cdot \vec{b})^{2}=|\vec{a}|^{2}|\vec{b}|^{2}$
$\Rightarrow 144=(4)^{2}|\vec{b}|^{2} $
$\Rightarrow 16|\vec{b}|^{2}=144$
$\Rightarrow | \vec{ b }|^{2}=3$
Hence, Assertion is false.
$(\vec{a} \times \vec{b})^{2}+(\vec{a} \cdot \vec{b})^{2}=|\vec{a} \times \vec{b}|^{2}+(\vec{a} \times \vec{b})^{2}$
$=(a b \sin \theta)^{2}+(a b \cos \theta)^{2}=\vec{a}^{2} \vec{b}^{2}$
$\Rightarrow (\vec{a} \times \vec{b})^{2}=\vec{a}^{2} \vec{b}^{2}-(\vec{a} \cdot \vec{b})^{2}$
Hence Reason is true