JEE Advanced 2016 Paper 1 · Mathematics · Vector Algebra · hard

Let vec a,vec b,vec c be unit vectors such that the angle between each pair is (π)/(3). If vec d=vec a×vec b+vec b×vec c+vec c×vec a, then the value…

Numerical answer

Let

\vec a,\vec b,\vec c

be unit vectors such that the angle between each pair is

\dfrac{\pi}{3}

. If

\vec d=\vec a\times\vec b+\vec b\times\vec c+\vec c\times\vec a

, then the value of

4\,|\vec d|^2

Show answer and solution

Answer: 3

Use the Gram matrix:

\vec a\cdot\vec a=1

and every cross dot product is

\tfrac12

. For two cross products,

(\vec a\times\vec b)\cdot(\vec c\times\vec d)=(\vec a\cdot\vec c)(\vec b\cdot\vec d)-(\vec a\cdot\vec d)(\vec b\cdot\vec c)

. Squares:

|\vec a\times\vec b|^2=1-\tfrac14=\tfrac34

, and likewise for the other two, giving

\tfrac94

in total. Cross terms:

(\vec a\times\vec b)\cdot(\vec b\times\vec c)=(\vec a\cdot\vec b)(\vec b\cdot\vec c)-(\vec a\cdot\vec c)(\vec b\cdot\vec b)=\tfrac14-\tfrac12=-\tfrac14

, and each of the three pairs gives the same value. So

|\vec d|^2=\tfrac94+2\left(3\cdot\left(-\tfrac14\right)\right)=\tfrac94-\tfrac32=\tfrac34

. Hence

4|\vec d|^2=3

. A numerical construction of the vectors with this Gram matrix reproduced

0.75

Original question written in the JEE Advanced 2016 Paper 1 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake