InterviewSolution
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InterviewSolution
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दो रेखाओं जिनके सदिश समीकरण निम्नलिखित हैं, के बीच कि न्यूनतम दुरी ज्ञात करें| (i) `vecr = (3-t)hati + (4+2t)hatj + (t-2)hatk` तथा (and) `vecr = (1+s)hati + (3s-7)hatj + (2s-2)hatk` |
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Answer» (i) दी हुई रेखाएँ हैं, `vecr = (3-t)hati + (4+2t)hatj + (t-2)hatk`.............(1) (1)से, `vecr =(3hati + 4hatj -2hatk)+t(-hati + 2hatj+hatk)` (1) से, `vecr =(3hati +4hatj -2hatk) + t(-hati + 2hatj +hatk)` `rArr vecr = veca_(1) + tvecb_(1)` जहाँ `veca_(1) = 3hati + 4hatj - 2hatk` तथा `vecb_(1) = -hati + 2hatj + hatk` (2) से, `vecr =(hati - 7hatj -2hatk)+s (hati + 3hatj + 2hatk)` `rArr vecr = veca_(2) + svecb_(2)` जहाँ `veca_(2) = hati-7hatj - 2hatk` or `vecb_(2) = hati + 3hatj + 2hatk` रेखाओं (1) तथा (2) के बीच कि न्यूनतम दुरी `d=|((veca_(2)-veca_(1))(vecb_(1)xx vecb_(2)))/(|vecb_(1) xx vecb_(2)|)|`..........(1) अब `veca_(2)-veca_(1) = (hati - 7hatj - 2hatk) -(3hati + 4hatj -2hatk) = -2hati-11hatj`........(2) `vecb_(1) xx vecb_(2) = |:(hati, hatj, hatk),(-1,2,1),(1,3,2):|=hati + 3hatj - 5hatk`..........(3) `|vecb_(1) xx vecb_(2)| = sqrt(1^(2) + 3^(2) + (-5)^(2))= sqrt(35)`..........(4) तथा `(veca_(2)-veca_(1)).(vecb_(1) xx vecb_(2)) = (-2hati-11hatj).(hati + 3hatj - 5hatk)` `=(-2) xx 1 +(-11) xx 3+0 =-35`............(5) `therefore d=|-35/sqrt(35)|=35/sqrt(35)=sqrt(35)` इकाई |
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