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प्रश्न
A(– 2, 3, 4), B(1, 1, 2) and C(4, –1, 0) are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear.
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उत्तर
We find the cartesian equations of the line AB.
The cartesian equations of the line passing through the points (x1, y1, z1) and (x2, y2, z2) are
`(x - x_1)/(x_2 - x_1) = (y - y_1)/(y_2 - y_1) = (z - z_1)/(z_2 - z_1)`
Here, (x1, y1, z1) ≡ (−2, 3, 4) and (x2, y2, z2) ≡ (1, 1, 2)
∴ The required cartesian equations of the line AB are
`(x - (-2))/(1 - (-2)) = (y - 3)/(1 - 3) = (z - 4)/(2 - 4)`
∴ `(x + 2)/(1 + 2) = (y - 3)/(-2) = (z - 4)/(-2)`
∴ `(x + 2)/(3) = (y - 3)/(-2) = (z - 4)/(-2)`
C = (4, −1, 0)
For x = 4, `(x + 2)/(3) = (4 + 2)/(3)` = 2
For y = –1, `(y - 3)/(-2) = (-1 - 3)/(-2)` = 2
For z = 0, `(z - 4)/(-2) = (0 - 4)/(-2)` = 2
∴ The coordinates of C satisfy the equations of line AB.
∴ C lies on the line passing through A and B.
Hence, A, B, C are collinear.
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