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Question
If the four points with positing vectors −2^i+^j+^k,^i+^j+^k,^j−^k and λ^j+^k are coplanar, then λ is equal to
If the four points with positing vectors −2^i+^j+^k,^i+^j+^k,^j−^k and λ^j+^k are coplanar, then λ is equal to
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Solution
The correct option is A 1
If four point (x1,y1,z1),(x2,y2,z2),(x3,y3,z3) and (x4,y4,z4) are coplanar, then
∣∣
∣∣x2−x1y2−y1z2−z1x3−x1y3−y1z3−z1x4−x1y4−y1z4−z1∣∣
∣∣=0
Now, ∣∣
∣∣30020−22λ−10∣∣
∣∣=0
⇒3(0+2λ−2)=0
⇒λ=1
If four point (x1,y1,z1),(x2,y2,z2),(x3,y3,z3) and (x4,y4,z4) are coplanar, then
∣∣ ∣∣x2−x1y2−y1z2−z1x3−x1y3−y1z3−z1x4−x1y4−y1z4−z1∣∣ ∣∣=0
Now,
∣∣
∣∣30020−22λ−10∣∣
∣∣=0
⇒3(0+2λ−2)=0
⇒λ=1
⇒λ=1
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