Verified Solution • 2025–26 Session moderate

QJEEM07Ma9030: JEE Main 2007 Physics - Class 11 JEEM07Ma9030 Solution: If and are unit vectors and is the acute angle between them,…

If u^\widehat u and v^\widehat v are unit vectors and θ\theta is the acute angle between them, then 2u^×3v^2\widehat u \times 3\widehat v is a unit vector for :

Multiple Choice Options

  • A no value of θ\theta
  • B exactly one value of θ\theta
  • C exactly two values of θ\theta
  • D more than two values of θ\theta

Step-by-Step Verified Solution

Given ∣2u^×3v^∣=1\left| {2\widehat u \times 3\widehat v} \right| = 1

and θ\theta is acute angle between u^\widehat u

and v^,  ∣u^∣=1,  ∣v^∣=1   \widehat v,\,\,\left| {\widehat u} \right| = 1,\,\,\left| {\widehat v} \right| = 1\,\,\,

⇒   6∣u^∣∣v^∣∣sin⁡θ∣=1\Rightarrow \,\,\,6\left| {\widehat u} \right|\left| {\widehat v} \right|\left| {\sin \theta } \right| = 1

⇒6∣sin⁡θ∣=1⇒sin⁡θ=16\Rightarrow 6\left| {\sin \theta } \right| = 1 \Rightarrow \sin \theta = {1 \over 6}

Hence, there is exactly one value of θ\theta

for which 2u^×3v^2\widehat u \times 3\widehat v is a unit vector.

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