Question

The vectors `lambdahat"i" + hat"j" + 2hat"k", hat"i" + lambdahat"j" - hat"k"` and `2hat"i" - hat"j" + lambdahat"k"` are coplanar if ______.

विकल्प

• λ = –2

• λ = 0

• λ = 1

• λ = – 1

Answer 1

The vectors `lambdahat"i" + hat"j" + 2hat"k", hat"i" + lambdahat"j" - hat"k"` and `2hat"i" - hat"j" + lambdahat"k"` are coplanar if λ = –2.

Explanation:

Let `vec"a" = lambdahat"i" + hat"j" + 2hat"kk"`

`vec"b" = hat"i" + lambdahat"j" - hat"k"`

`vec"c" = 2hat"i" - hat"j" + lambdahat"k"`

If `vec"a", vec"b", vec"c"` are coplanar, then

`vec"a" * (vec"b" xx vec"c")` = 0

∴ `|(lambda, 1, 2),(1, lambda, -1),(2, -1, lambda)|` = 0

⇒ λ(l² – 1) – 1 (λ + 2) + 2(–1 – 2λ) = 0

⇒ λ³ – λ – λ – 2 – 2 – 4λ = 0

⇒ λ³ – 6λ – 4 = 0

⇒ (λ + 2)(λ² – 2λ – 2) = 0

⇒ λ = – 2 or λ² – 2λ – 2 = 0

⇒ `lambda = (2 +- sqrt(4 + 8))/2`

⇒ `lambda = (2 +- 2sqrt(3))/2`

∴ `lambda = - 2` or `lambda = 1 +- sqrt(3)`.