Let the volume of a parallelopiped whose coterminous edges are gi… - Vidyadip

MCQ

Let the volume of a parallelopiped whose coterminous edges are given by $\overrightarrow{\mathrm{u}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}, \overrightarrow{\mathrm{v}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}} $ and $\overrightarrow{\mathrm{w}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ be $1\; cu.$ unit. If $\theta$ be the angle between the edges $\overrightarrow{\mathrm{u}}$ and $\overrightarrow{\mathrm{w}},$ then $\cos \theta$ can be

✓
$\frac{7}{6 \sqrt{3}}$
B
$\frac{5}{7}$
C
$\frac{7}{6 \sqrt{6}}$
D
$\frac{5}{3 \sqrt{3}}$

✓

Answer

Correct option: A.

$\frac{7}{6 \sqrt{3}}$

a
$\left|\begin{array}{lll}{1} & {1} & {\lambda} \\ {1} & {1} & {3} \\ {2} & {1} & {1}\end{array}\right|=1 \Rightarrow \lambda=2,4$

$\mathrm{Now}, \quad \cos \theta=\frac{\overrightarrow{\mathrm{u}} \cdot \overrightarrow{\mathrm{w}}}{|\overrightarrow{\mathrm{u}}||\overrightarrow{\mathrm{w}}|}$

$=\frac{5}{\sqrt{6} \sqrt{6}}$ or $\frac{7}{\sqrt{6} \sqrt{18}}=\frac{5}{6}$ or $\frac{7}{6 \sqrt{3}}$

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