Question

If `overline(a),overline(b),overline(c)` are non-coplanar vectors and λ is a real number then `[lambda(overline(a)+overline(b))lambda^2overline(b)  lambda overline(c)]=[overline(a)  overline(b)+overline(c)  overline(b)]` for ______.

Options

• exactly three values of λ

• exactly two values of λ

• exactly one value of λ

• no value of λ

Answer 1

If `overline(a),overline(b),overline(c)` are non-coplanar vectors and λ is a real number then `[lambda(overline(a)+overline(b))lambda^2overline(b)  lambda overline(c)]=[overline(a)  overline(b)+overline(c)  overline(b)]` for no value of λ.

Explanation:

`[lambda(overline(a)+overline(b))lambda^2overline(b)  lambda overline(c)]=[overline(a)  overline(b)+overline(c)  overline(b)]`

`=> lambda^4[overline(a)+overline(b)  overline(b)  overline(c)]=[overline(a)  overline(b)+overline(c)  overline(b)]`

`=>lambda^4{[overline(a)  overline(b)  overline(c)]+[overline(b) overline(b)  overline(c)]}={[overline(a)  overline(b)  overline(b)]+[overline(a)  overline(c)  overline(b)]}`

`=>lambda^4[overline(a)  overline(b)  overline(c)]=-[overline(a)  overline(b)  overline(c)]`

`=>(lambda^4+1)[overline(a)  overline(b)  overline(c)]=0`

But, `[overline(a)  overline(b)  overline(c)] ne0`

`thereforelambda^4+1=0`

This is not true for any real value of λ.