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Show that the points `A(1,-2,-8),B(5,0,-2)a n dC(1,3,7)`are collinear, and find the ratio in which `B`divides `A Cdot` |
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Answer» The position vectors of A, B and C are `(hati - 2hatj - 8hatk), (5hati-2hatk)` and `(11hati + 3hatj + 7hatk)` respectively. `:. vec(AB) =` (position vector of B) - (position vector of A) `= (5hati - 2hatk( - (hati - 2hatj +8hatk) = (4hati + 2hatj + 6hatk)`. `vec(BC) =` (position of C) - (position vector of B) `= (11hati + 3hatj + 7hatk) - (5hati - 2hatk) = (6hati+3hatj +9hatk)`, and `vec(AC) =` (position vector of C) - (position vector of A) `(11hati + 3hatj + 7hatk) - (-hati - 2hatj - 8hatk) = (10 hati + 5hatj + 15 hatk)`. Now, `vec(AB) = (4hati + 2hatj + 6hatk) = 2(2hati + hatj + 3hatk)`. `= 2/10 (6 hati + 3hatj + 9hatk) = 2/3 vec(BC)`. `:. (|vec(AB)|)/(|vec(BC)|) = 2/3` Hence, B divides AC in the ratio `2:3`. |
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