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\(\rm \left [\vec a \times \vec b,\ \vec b \times \vec c,\ \vec c \times \vec a \right]\) = 64 எனில் \(\rm \left [\vec a\ \vec b\ \vec c \right]\) ன் மதிப்பு

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TNUSRB SI 2010 Official Paper (Held on 04 July 2010)
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  1. 32
  2. 8
  3. 128
  4. 0

Answer (Detailed Solution Below)

Option 2 : 8
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விளக்கம்:
மூன்று திசையன்களின் ஸ்கேலார் ட்ரிபிள் பெருக்கம் → a a ⃗ , → b b ⃗ , மற்றும் → சிc⃗ _ [ → a , → b , → c ] [ a ⃗ , b என குறிக்கப்படுகிறது⃗ , c ⃗ ] மற்றும் வரையறுக்கப்பட்டது[
→ a , → b , → c ] = → a ⋅ ( → b ×→ c ) = → b ⋅ ( → c × → a ) = → c ⋅ ( → a × → b )
[ a ⃗ , b ⃗ , c ⃗ ] = a ⃗ ⋅ ( b ⃗× c ⃗ ) = b ⃗ ⋅ ( c ⃗ × a ⃗ ) = c ⃗⋅ ( a ⃗ × b ⃗ )
 
இந்த தயாரிப்பு 3x3 மேட்ரிக்ஸின் நிர்ணயிப்பிற்கு சமம், அதன் வரிசைகள் (அல்லது நெடுவரிசைகள்) திசையன்களின் கூறுகளாகும் → a a ⃗ ,→ b b ⃗ , மற்றும் → c c ⃗ .
 
கொடுக்கப்பட்ட [ → a × → b, → b × → c , → c × → a ] = 64 [ a ⃗ × b ⃗, b ⃗ × c ⃗ , c ⃗ × a ⃗ ] = 64 , இதை [ → என மாற்றி எழுத சுழற்சி வரிசைமாற்றங்களின் பண்பைப் பயன்படுத்தலாம்a , → b , → c ] 2 = 64 [ a ⃗ , b ⃗ , c ⃗] 2 = 64 . எனவே, [ → a , → b , → c ] = ± 8 [ a ⃗ ,b ⃗ , c ⃗ ] = ± 8 .
 
இருப்பினும், வழங்கப்பட்ட விருப்பங்களில் − 8 - 8 இல்லை , எனவே நாம் முடிவு செய்யலாம் [ → a , → b ,→ c ] = 8 [ a ⃗ , b ⃗ , c ⃗ ] = 8 , இது விருப்பம் 2 க்கு ஒத்திருக்கிறது.
 
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