JEE Advanced 2019 Paper 2 · Mathematics · Vector Algebra · medium

Let vec a,vec b,vec c be unit vectors such that vec a·vec b=vec b·vec c=vec c·vec a=1/2. The volume of the parallelepiped whose coterminous edges are…

Numerical answer

Let

\vec a,\vec b,\vec c

be unit vectors such that

\vec a\cdot\vec b=\vec b\cdot\vec c=\vec c\cdot\vec a=\dfrac12

. The volume of the parallelepiped whose coterminous edges are

\vec a+\vec b

\vec b+\vec c

and

\vec c+\vec a

is ______. (Give the answer correct to two decimal places.)

Show answer and solution

Answer: 1.41

First,

[\vec a+\vec b,\ \vec b+\vec c,\ \vec c+\vec a]=2[\vec a\ \vec b\ \vec c]

, because the determinant of the coefficient matrix

\begin{pmatrix}1&1&0\\0&1&1\\1&0&1\end{pmatrix}

equals

2

. Next,

[\vec a\ \vec b\ \vec c]^2

equals the determinant of the Gram matrix

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

, whose determinant is

1+2\cdot\frac18-3\cdot\frac14=\frac12

. So

|[\vec a\ \vec b\ \vec c]|=\frac1{\sqrt2}

and the required volume is

2\cdot\frac{1}{\sqrt2}=\sqrt2\approx1.41

. An explicit numerical construction of three such unit vectors reproduces

1.4142

Original question written in the JEE Advanced 2019 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