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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 ______.
Options
λ = –2
λ = 0
λ = 1
λ = – 1
Solution
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)`.
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