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Question
If the points (–1, –1, 2), (2, m, 5) and (3,11, 6) are collinear, find the value of m.
Solution
Let the given points be A(–1, –1, 2), B(2, m, 5) and C(3, 11, 6).
Then `vec"AB" = (2 + 1)hat"i" + ("m" + 1)hat"j" + (5 - 2)hat"k"`
= `3hat"i" + ("m" + 1)hat"j" + 3hat"k"`
And `vec"AC" = (3 + 1)hat"i" + (11 + 1)hat"j" + (6 - 2)hat"k"`
= `4hat"i" + 12hat"j" + 4hat"k"`.
Since A, B, C, are collinear
We have `vec"AB" = lambda vec"AC"`
i.e., `(3hat"i" + ("m" + 1)hat"j" + 3hat"k") = lambda(4hat"i" + 12hat"j" + 4hat"k")`
⇒ 3 = `4lambda` and m + 1 = `12lambda`
Therefore m = 8.
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