Question

# Show that the points (2, −1, 3), (4, 3, 1) and (3, 1, 2) are collinear.

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Hint: Here we go through by finding the direction ratios between the points, because we know that the points are collinear if their direction ratios are proportional.

Given points are (2, −1, 3), (4, 3, 1) and (3, 1, 2).

Now we let the names of points as A (2, −1, 3), B (4, 3, 1) and C (3, 1, 2).
We know that three points A, B, C are collinear if direction ratios of AB and BC are proportional.

Now we have to find the direction ratio of AB,
As we know the formula of finding the direction ratio between two point $({x_1},{y_1},{z_1})$ and $({x_2},{y_2},{z_2})$ is $({x_2} - {x_1},{y_2} - {y_1},{z_2} - {z_1})$
Therefore the direction ratio of AB is (4-2, 3-(-1), 1-3) i.e. (2, 4, -2).
Similarly we find the direction ratio of BC (3-4, 1-3, 2-1) i.e. (-1, -2, 1).

Let the direction ratio of AB (2, 4, -2) as $({a_1},{b_1},{c_1})$ and direction ratio of BC (-1, -2, 1) as $({a_2},{b_2},{c_2})$
$\therefore {a_1} = 2,{\text{ }}{b_1} = 4,{\text{ }}{c_1} = - 2$ And ${a_2} = - 1,{\text{ }}{b_2} = - 2,{\text{ }}{c_2} = 1$
And now we check whether these points are in proportion or not.

$\therefore \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}} = \dfrac{{ - 2}}{1}$
Therefore A, B, C are collinear.

Note: Whenever we face such a type of question for finding the collinearly of a point whether it is collinear or not. We have to simply find the direction ratio of that point if their direction ratio is in proportion we can say that the points are collinear. The second method is just to find the value of the determinant of the following point if the result is zero then we can say that lines are collinear.