सदिश `hati + hatj + hatk` का, सदिशों `2hati + 4hatj - 5hatk` और के योगफल की दिशा में मात्रक संदिश के साथ अदिश गुणनफल 1 के बरोबर है तो `lambda` का मान ज्ञात कीजिए।

यदि A (-hati + 3hatj + 2hatk) , B (-4hati + 2hatj - 2hatk) तथा C(5hati + (lambda +1)hatj + mu hatk) संरेखीय है तब :

Find the shortest distance between the lines: (i) vec(r) = 3 hati + 8 hat(j) + 3 hatk + lambda (3 hati - hatj + hatk) and vec(r) = - 3 hat(i) - 7 hatj + 6 hatk + mu (-3 hati + 2 hatj + 4 hatk ) (ii) ( hati - hatj + 2 hatk) + lambda ( -2 hati + hatj + 3 hatk ) and (2 hati + 3 hatj - hatk) + mu (3 hati - 2 hatj + 2 hatk). (iii) vec(r) = (hati + 2 hatj + 3 hatk) + lambda ( hati - 3 hatj + 2 hatk ) and vec(r) = (4 hati + 5 hatj + 6 hatk) + mu (2 hati + 3 hatj + hatk) .

Find the angle between the following pairs of lines : (i) vec(r) = 2 hati - 5 hatj + hatk + lambda (3 hati + 2 hatj + 6 hatk ) and vec(r) = 7 hati - 6 hatk + mu (hati + 2 hatj + 2 hatk) (ii) vec(r) = 3 hati + hatj - 2 hatk + lambda (hati - hatj - 2 hatk ) and vec(r) = 2 hati - hatj - 56 hatk + mu (3 hati - 5 hatj - 4 hatk) .

Show that the lines : vec(r) = 3 hati + 2 hatj - 4 hatk + lambda (hati + 2 hatj + 2 hatk) and vec(r) = 5 hati - 2 hatj + mu (3 hati + 2 hatj + 6 hatk) (ii) vec(r) = (hati + hatj - hatk) + lambda (3 hati - hatj) . and vec(r) = (4 hati - hatk) + mu (2 hati + 3 hatk) are intersecting. Hence, find their point of intersection.

सदिशों का अदिश गुणनफल (Scalar Product of Vectors)|तीन विमाओं में दो सदिशों का अदिश गुणनफल (Product of Two Vectors in Three Dimensions)|भौतिकी में अदिश गुणनफल के उदाहरण (Examples of Scalar Product in Physics)|प्रश्न (Question)|सदिशों का सदिश गुणनफल (Vector Product of Vectors)|तीन विमाओं में दो सदिशों का सदिश गुणनफल (Cross Product of Two Vectors in Three Dimensions)|प्रश्न (Question)