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 $\vec{u} \cdot \vec{w}=1, \vec{v} \cdot \vec{w}=1, \vec{w} \cdot \vec{w}=4$ If the volume of the parallelopiped, 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 ____.

JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · Vector Algebra

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 \cdot w = 1, v \cdot w = 1, w \cdot w = 4

If the volume of the parallelopiped, whose adjacent sides are represented by the vectors

u, v

and

w

, is

2

, then the value of

∣3 u + 5 v ∣

is ____.

Answer: 7

Given:

∣ u ∣ = 1, ∣ v ∣ = 1, u \cdot v \neq = 0, u \cdot w = 1, v \cdot w = 1, ∣ w ∣^{2} = 4

∣ w ∣ = 2

. The volume is

[u v w] = 2

, so

[u v w]^{2} = 2

. Using the formula

[u v w]^{2} = u \cdot u v \cdot u w \cdot u u \cdot v v \cdot v w \cdot v u \cdot w v \cdot w w \cdot w = 1 u \cdot v 1 u \cdot v 11 114 = 2

. Expanding the determinant and solving gives

u \cdot v = \frac{1}{2}

. Therefore,

∣3 u + 5 v ∣ = 9∣ u ∣^{2} + 25∣ v ∣^{2} + 30 u \cdot v = 9 + 25 + 30 (\frac{1}{2}) = 49 = 7

JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · Integer answer