JEE Advanced 2024 Paper 2 · Mathematics · Vector Algebra · easy

Let vec a,vec b be unit vectors with angle (π)/(3) between them, and let vec c=2(vec a×vec b)+3vec a-vec b. The value of 4 [vec a vec b vec c]^2 is

Numerical answer

Let

\vec a,\vec b

be unit vectors with angle

\dfrac\pi3

between them, and let

\vec c=2(\vec a\times\vec b)+3\vec a-\vec b

. The value of

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

Show answer and solution

Answer: 9

The scalar triple product

[\vec a\ \vec b\ \vec c]=\vec c\cdot(\vec a\times\vec b)

. The components

3\vec a

and

-\vec b

\vec c

lie in the plane of

\vec a,\vec b

and are perpendicular to

\vec a\times\vec b

, so they contribute

0

. Thus

[\vec a\ \vec b\ \vec c]=2|\vec a\times\vec b|^2=2\sin^2\dfrac\pi3=2\cdot\dfrac34=\dfrac32

. Then

4[\vec a\ \vec b\ \vec c]^2=4\cdot\dfrac94=9

. A direct numerical computation with

\vec a=(1,0,0)

\vec b=(\tfrac12,\tfrac{\sqrt3}2,0)

gives the same value.

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