NIMCET 2014 Question Paper with solutions

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Question : 19 of 120

Marks: +1, -0

A=

4i↖{\^}+3j↖{\^}+k↖{\^}

, B =

2i↖{\^}-j↖{\^}+2k↖{\^}

, then the unit vector

N↖{\^}

perpendicular to vector A and B such that A , B >

N↖{\^}

form a right handed system , is

$1/√{185}$ $[7i↖{\^}-6j↖{\^}-10k↖{\^}]$
$1/7 [6i↖{\^}+2j↖{\^}+3k↖{\^}]$
$1/√21 [2i↖{\^}+4j↖{\^}-k↖{\^}]$
$1/√21 [-2i↖{\^}-4j↖{\^}+k↖{\^}]$

Solution:

We have
A=

4i↖{\^}+3j↖{\^}+k↖{\^}

B =

2i↖{\^}-j↖{\^}+2k↖{\^}

N is perpendicular to both A and B.
i.e.

N↖{\^}

= λ (A × B)

N↖{\^}

= λ

[\table i↖{\^},j↖{\^},k↖{\^};4,3,1;2,-1,2]

= λ

[i↖{\^}(6+1)-j↖{\^}(8-2)+k↖{\^}(-4-6)]

= λ

[7i↖{\^}-6j↖{\^}-10k↖{\^}]

∴

|N↖{\^}|

= 1

|N↖{\^}|

= λ

√{185}

= 1
⇒ λ =

1/{√185}

∴

N↖{\^}

=

1/√{185}

(7i↖{\^}-6j↖{\^}-10k↖{\^})

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