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प्रश्न
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`
उत्तर
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`
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