Let u→ , v→ and w→ be vectors in three-dimensional space, where u→ and v→ are unit vectors which are not perpendicular to each other and u→.w→=1, v→.w→=1, w→.w→=4If the volume of the parallelepiped, whose adjacent sides are represented by the vectors u→, v→ and w→ is 2, then the value of |3u→+5v→| is __________.

Volume of parallelepiped =[u→ v→ w→]=2Now, [u→ v→ w→]2=|u→.u→u→.v→u→.w→v→.u→v→.v→v→.w→w→.u→w→.v→w→.w→|∴ |1u→.v→1v→.u→11114|=2∴ u→.v→ = 12 ⇒ |3u→+5v→|2=9|u→|2+25|v→|2+30u→.v→ =34+15=49

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Let $\vec{u}, \vec{v}$ and $\vec{w}$ be vectors in three-dimensional space, where $\vec{u}$ and $\vec{v}$ are unit vectors which are not perpendicular to each other and $\vec{u} . \vec{w} = 1, \vec{v} . \vec{w} = 1, \vec{w} . \vec{w} = 4$
If the volume of the parallelepiped, whose adjacent sides are represented by the vectors $\vec{u}$ , $\vec{v}$ and $\vec{w}$ is $\sqrt{2},$ then the value of $| 3 \vec{u} + 5 \vec{v} |$ is __________.

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Detailed Solution

Volume of parallelepiped $= [\vec{u} \vec{v} \vec{w}] = \sqrt{2}$
Now, ${[\vec{u} \vec{v} \vec{w}]}^{2} = | \begin{matrix} \vec{u} . \vec{u} & \vec{u} . \vec{v} & \vec{u} . \vec{w} \\ \vec{v} . \vec{u} & \vec{v} . \vec{v} & \vec{v} . \vec{w} \\ \vec{w} . \vec{u} & \vec{w} . \vec{v} & \vec{w} . \vec{w} \end{matrix} |$
$∴ | \begin{matrix} 1 & \vec{u} . \vec{v} & 1 \\ \vec{v} . \vec{u} & 1 & 1 \\ 1 & 1 & 4 \end{matrix} | = 2$

$∴ \vec{u} . \vec{v} = \frac{1}{2} \Rightarrow {| 3 \vec{u} + 5 \vec{v} |}^{2} = 9 {| \vec{u} |}^{2} + 25 {| \vec{v} |}^{2} + 30 \vec{u} . \vec{v} = 34 + 15 = 49$