JEE Advanced 2023 Paper 2 · Mathematics · Vector Algebra · medium

Let vec u,vec v,vec w be unit vectors such that the angle between every two of them is 60^°. The volume of the parallelepiped whose coterminous edges…

Single correct option

Let

\vec u,\vec v,\vec w

be unit vectors such that the angle between every two of them is

60^\circ

. The volume of the parallelepiped whose coterminous edges are

\vec u+\vec v,\ \vec v+\vec w,\ \vec w+\vec u

Show answer and solution

Answer: (C) $\sqrt2$

The Gram matrix of

\vec u,\vec v,\vec w

G=\begin{pmatrix}1&\frac12&\frac12\\ \frac12&1&\frac12\\ \frac12&\frac12&1\end{pmatrix}

with

\det G=\frac12

, and

[\vec u\,\vec v\,\vec w]^2=\det G

, so

|[\vec u\,\vec v\,\vec w]|=\frac1{\sqrt2}

. Now

[\vec u+\vec v,\vec v+\vec w,\vec w+\vec u]=2[\vec u\,\vec v\,\vec w]

(the coefficient matrix has determinant

2

). Volume

=\frac{2}{\sqrt2}=\sqrt2

. A numerical construction with a Cholesky factor of

G

confirms this. The option

\frac1{\sqrt2}

forgets the factor

2

Original question written in the JEE Advanced 2023 Paper 2 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake