Question
L1 and L2 are two lines whose vector equations are L1:r⃗ =λ((cosθ+3√)i^+(2√sinθ)j^+(cosθ−3√)k^)L2:r⃗ =μ(ai^+bj^+ck^), where λ and μ are scalars and α is the acute angle between L1 andL2. If the angle ′α′ is independent of θ then the value of ′α′ is
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[a] Both the lines pass through origin. Line L1 is parallel to the vector V−→2=ai^+bj^+ck^ ∴cosα=V1.−→−V2−→−|V1−→−||V2−→−| =a(csoθ+3√)+(b2√)sinθ+c(cosθ−3√)a2+b2+c2−−−−−−−−−−√(cosθ+3√)2+2sin2θ+(cosθ−3√)2−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√=(a+c)cosθ+b3√)+sinθ+(a−c−3√)a2+b2+c2−−−−−−−−−−√2+6−−−−√ In order that cosα is independent ofθ, we get a+c=0 and b=0 ∴cosα=2a3√a2√22√=3√2⇒α=π6
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Question Text | L1 and L2 are two lines whose vector equations are L1:r⃗ =λ((cosθ+3√)i^+(2√sinθ)j^+(cosθ−3√)k^)L2:r⃗ =μ(ai^+bj^+ck^), where λ and μ are scalars and α is the acute angle between L1 andL2. If the angle ′α′ is independent of θ then the value of ′α′ is |
Answer Type | Text solution:1 |
Upvotes | 151 |