If →a=^i+2^j+4^k,→b=^i+λ^j+4^k,→c=2^i+4^j+(λ2−1)^k, are coplanar vectors and →a,→c are not parallel, then |→a×→c| is equal to
A
5√5
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B
2√5
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C
3√5
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D
25
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Solution
The correct option is A5√5 Given: →a,→b,→c are coplanar. ⇒∣∣
∣∣1241λ424λ2−1∣∣
∣∣=0 ⇒1(λ(λ2−1)−16)−2(λ2−1−8)+4(4−2λ)=0 ⇒λ3−2λ2−9λ+18=0 ⇒(λ−2)(λ2−9)=0 ⇒λ=2 or λ=±3
But for λ=±3,→a is parallel to →c.
So, λ=2 ∴→a×→c=∣∣
∣
∣∣^i^j^k124243∣∣
∣
∣∣ ⇒→a×→c=^i(6−16)−^j(3−8)+^k(4−4) ⇒→a×→c=−10^i+5^j
Now, |→a×→c|=√100+25=5√5