Let u- v and w be three 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 - 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 -3u-5v- is

Given: nbsp; [u v w ]=√(2) ⇒ [ u v w ]^2 ⇒u.u amp;u.v nbsp; amp;u.w nbsp; v.u amp;v.v nbsp; amp;v.w nbsp; w.u amp;w.v nbsp; amp; w.w = ...

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Byju's Answer

Standard XII

Mathematics

Applications of Cross Product

Let u, v and ...

Question

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

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Solution

Given: $[\to u \to v \to w] = \sqrt{2}$
$\Rightarrow {[\to u \to v \to w]}^{2}$
$\Rightarrow ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to u . \to u & \to u . \to v & \to u . \to w \to v . \to u & \to v . \to v & \to v . \to w \to w . \to u & \to w . \to v & \to w . \to w \end{matrix} ∣ ∣ ∣ ∣ ∣ = 2$

Let $\to u \cdot \to v = k$ . Then
$∣ ∣ ∣ ∣ \begin{matrix} 1 & k & 1 k & 1 & 1 1 & 1 & k \end{matrix} ∣ ∣ ∣ ∣ = 2$
$\Rightarrow 4 k^{2} - 2 k = 0$
$\Rightarrow \to u . \to v = \frac{1}{2}$ or $\to u . \to v = 0$ (rejected)
$∴ \to u . \to v = \frac{1}{2}$
Now,
$| 3 \to u + 5 \to v |^{2} = 9 + 25 + 30 \times \frac{1}{2} = 49$
$∴ | 3 \to u + 5 \to v | = 7$

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