Question

Practice set 1( mathamatics)
Important for jee mains exam ​

Answer 1

Answer:

Option (a) [a b c ] r

Step-by-step explanation:

a,b,c are three non - coplaner vectors

∵ we know that a, b and c are non coplanar vectors; then any vector r can be expressed as linear combination: xa+yb+zc

• r = xa + yb + zc ________(1)

multipling by (a×b) in LHS and RHS

r•(a×b) = xa(a×b) + yb (a×b) + z c•(a×b)

• r•(a×b) = 0 + 0 + [a b c]
• r•(a×b) = z[a b c]
• z = r• (a×b) / [a b c]

Multipling by (b×c)in LHS and RHS

• r•(b×c) = x a•(b×c)
• x = r • ( b×c) / [a b c]

Similarly multipling by (c×a]

• r• ( c×a) = y b•(c × a)
• r • ( c× a) = y [ a b c ]
• y = r •(c × a) / [a b c ]

from (1) r = xa+ yb + zc

putting the value of x , y and z

we get ,

r = ar• ( b× c) / [a b c ] + b r • (c×a)/[a b c] + c r • ( a× b)/[ a b c ]

r = a [ r b c ] + b [ r c a ] + c [ r a b ] / [ a b c ]

Taking [a b c ] LCM

(r) [ a b c ] = a [ r b c ] + b [ r c a ] + c [ r a b ]

a [ r b c ] + b [ r c. a ] + c [ r a b ] = r [ a b c ] Answer