Question

If the vectors, `vecp = (a + 1)hati + ahatj + ahatk, vecq = ahati + (a + 1)hatj + ahatk` and `vecr = ahati + ahatj + (a + 1)hatk  (a ∈ R)` are coplanar and `3(vecp.vecq)^2 - λ|vecr xx vecq|^2` = 0, then the value of λ is ______.

Options

• 0

• 1

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• 3

Answer 1

If the vectors, `vecp = (a + 1)hati + ahatj + ahatk, vecq = ahati + (a + 1)hatj + ahatk` and `vecr = ahati + ahatj + (a + 1)hatk  (a ∈ R)` are coplanar and `3(vecp.vecq)^2 - λ|vecr xx vecq|^2` = 0, then the value of λ is 1.

Explanation:

Since, `vecp, vecq` and `vecr` are coplanar.

`|(a + 1, a, a),(a, a + 1, a),(a, a, a + 1)|` = 0

`\implies` 3a + 1 = 0

`\implies` a = `-1/3`

The given vectors

`vecp = 2/3hati - 1/3hatj - 1/3hatk = 1/3(2hati - hatj - hatk)`

`vecq = 1/3(-hati + 2hatj - hatk)`

`vecr = 1/3(-hati - hatj + 2hatk)`

Now, `vecp.vecq = 1/9(-2 - 2 + 1) = -1/3`

`vecr xx vecq = 1/9|(hati, hatj, hatk),(-1, 2, -1),(-1, -1, 2)|`

= `1/9(hati(4 - 1) - hatj(-2 - 1) + hatk(1 + 2)`

= `1/9(3hati + 3hatj + 3hatk)`

= `(hati + hatj + hatk)/3`

`|vecr xx vecq| = 1/3sqrt(3)`

`\implies |vecr xx vecq|^2 = 1/3`

`3(vecp.vecq)^2 - λ|vecr xx vecq|^2` = 0

`\implies 3. 1/9 - λ. 1/3` = 0

`\implies` λ = 1