Vector Algebra
Vector Algebra
nta_abhyas_2025
Grade 12
Question:
Let the volume of a parallelepiped whose coterminous edges are given by $\vec{u} = \vec{i} + \vec{j} + \vec{k}$, $\vec{v} = \vec{i} + \vec{j} + 3\vec{k}$ and $\vec{w} = 2\vec{i} + \vec{j} + \vec{k}$ be 1 cu. unit. If $\theta$ is the angle between the edges $\vec{u}$ and $\vec{w}$, then the value of $\cos \theta$ can be
\frac{2}{\sqrt{6}}
\frac{2}{\sqrt{3}}
\frac{3}{\sqrt{6}}
\frac{5}{\sqrt{3}}
Step-by-Step Solution
Key Concept: Using determinants to find eigenvalues and computing angles between vectors via the dot product formula
Given the determinant form with vectors, we compute the characteristic equation $\lambda + 3 = \pm 1$ and $\lambda = 2$ or $\lambda = 4$. For $\lambda = 4$, we calculate $\cos\theta = \frac{2\sqrt{11}}{\sqrt{6} \cdot \sqrt{33}} = \frac{\sqrt{2}}{3}$. The angle $\theta = \cos^{-1}\left(\frac{\sqrt{2}}{3}\right)$ gives the final answer.
Correct Answer: 4