Question

If three points are on the same line, then they are collinear. Points X, Y, and Z are on line m.

Answer 1

So X,Y,Z are collinear

Answer 2

Condition for collinearity of three points on a line.

Explanation:

We are given the following information in the question:

Point X, Y and Z lies on a line m.

We have to show that these points are collinear. Three points on a point are collinear if the area of triangle formed by these three points is zero.

We make the following assumptions.

Let
`(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3),` be the coordinates of point X, Y and Z respectively.

Then the area of triangle formed by these three points is given by:

`(1)/(2)\left[\begin{array}{ccc}x_1&y_1&z_1\\x_2&y_2&z_2\\x_3&y_3&z_3\end{array}\right] }`

Equating area of triangle to zero,

`\left[\begin{array}{ccc}x_1&y_1&z_1\\x_2&y_2&z_2\\x_3&y_3&z_3\end{array}\right] } = 0`

Hence, the above condition is the condition foe collinearity of three points on a given line.