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.
$(\vec{a}\times \vec{b}{)}^2+(\vec{a}\cdot \vec{b}{)}^2=|\vec{a}\times \vec{b}{|}^2+(\vec{a}\times \vec{b}{)}^2$
$=(ab\sin \theta {)}^2+(ab\cos \theta {)}^2={\vec{a}}^2{\vec{b}}^2$
$\Rightarrow (\vec{a}\times \vec{b}{)}^2={\vec{a}}^2{\vec{b}}^2-(\vec{a}\cdot \vec{b}{)}^2$
Hence Reason is true