Mathematics Part-II

Book: Mathematics Part-II

Chapter: 11. Three Dimensional Geometry

Subject: Maths - Class 12th

Q. No. 1 of Miscellaneous Exercise

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1

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, –1), (4, 3, –1).

Let OA be the line joining the origin (0,0,0) and the point A(2,1,1).

Let BC be the line joining the points B(3,5,−1) and C(4,3,−1)


Direction ratios of OA = (a1, b1, c1) ≡ [(2 - 0), (1 - 0), (1 - 0)] ≡ (2,1,1)


Direction ratios of BC = (a2, b2, c2) ≡ [(4 - 3), (3 - 5), (-1 + 1)]


≡ (1, -2, 0)


Given-


OA is to BC


we have to prove that -


a1a2 + b1b2 + c1c2 = 0


L.H.S = a1a2 + b1b2 + c1c2 = 2 × 1 + 1 × (−2) + 1 × 0 = 2 - 2 = 0


R.H.S = 0


Thus, L.H.S = R.H.S ….PROVED


Hence OA is to BC.


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Chapter Exercises

More Exercise Questions

6

If the lines and are perpendicular, find the value of k.