Question

The value of λ for which the vectors `3hat"i" - 6hat"j" + hat"k"` and `2hat"i" - 4hat"j" + lambdahat"k"` are parallel is ______.

पर्याय

• `2/3`

• `3/2`

• `5/2`

• `2/5`

Answer 1

The value of λ for which the vectors `3hat"i" - 6hat"j" + hat"k"` and `2hat"i" - 4hat"j" + lambdahat"k"` are parallel is `2/3`.

Explanation:

Let `vec"a" = 3hat"i" - 6hat"j" + hat"k"`

`vec"b" = 2hat"i" - 4hat"j" + lambdahat"k"`

Since the given vectors are parallel,

∴ Angle between them is 0°

So `vec"a"*vec"b" = |vec"a"||vec"b"| cos 0`

⇒ `(3hat"i" - 6hat"j" + hat"k")*(2hat"i" - 4hat"j" + lambdahat"k") = |3hat"i" - 6hat"j" + hat"k"| |2hat"i" - 4hat"j" + lambdahat"k"|`

`6 + 24 + lambda = sqrt(9 + 36 + 1) * sqrt(4 + 16 + lambda^2)`

`30 + lambda = sqrt(46) * sqrt(20 + lambda^2)`

Squaring both sides, we get

900 + λ² + 60λ = 46(20 + λ²)

⇒ 900 + λ² + 60λ = 920 + 46λ²

⇒ λ² – 46λ² + 60λ + 900 – 920 = 0

⇒ – 45λ² + 60λ – 20 = 0

⇒ 9λ² – 12λ + 4 = 0

⇒ (3λ – 2)² = 0

⇒ 3λ – 2 = 0

⇒ 3λ = 2

∴  λ = `2/3`

Alternate method:

Let `vec"a" = "a"_1hat"i" + "a"_2hat"j" + "a"_3hat"k"` 

And `vec"b" = "b"_1hat"i" + "b"_2hat"j" + "b"_3hat"k"`

If `vec"a" | | vec"b"`

∴ `"a"_1/"b"_1 = "a"_2/"b"_2 = "a"_3/"b"_3`

⇒ `3/2 = (-6)/(-4) = 1/lambda`

⇒ `1/lambda = 3/2`

⇒ λ = `2/3`