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Question

# Lines OA,OB are drawn from O with direction cosines proportional to (1,−2,−1),(3,−2,3). Find the direction cosines of the normal to the plane AOB

A
±429±329±229
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B
±229±329±229
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C
±829±629±229
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D
±829±329±229
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Solution

## The correct option is A ⟨±4√29±3√29±−2√29⟩Let ax+by+cz+d=0 be the plane, then O(0,0,0);A(1,−2,−1);B(3,−2,3)⇒d=0 and a−2b−c=0 also 3a−2b+3c=0Putting c=(a−2b), we get 6a=8ba=43b∴a=43b,b=b and c=−2b3And the plane is b(43x+y−23z)=04x+3y−2z=0The normal ±(→n=4^i+3^j−2^k)^n=±(4√29^i+3√29^j+2√29^k)D.C' s of normal vector ⟨±4√29±3√29±−2√29⟩

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