If the line is passing through the points \((x_1, y_1, z_1)\) and has direction cosines l, m, n of the line, then which of the following is the cartesian equation of the line?

(a) \(\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}\)

(b) \(\frac{x-x_1}{n}=\frac{y-y_1}{m}=\frac{z-z_1}{l}\)

(c) \(\frac{x+x_1}{n}=\frac{y+y_1}{m}=\frac{z-z_1}{l}\)

(d) \(\frac{x+x_1}{l}=\frac{y+y_1}{m}=\frac{z+z_1}{n}\)

The question was posed to me in an international level competition.

My question is based upon Three Dimensional Geometry in division Three Dimensional Geometry of Mathematics – Class 12

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The correct choice is (a) \(\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}\)

Explanation: If the line is passing through the points (x1, y1, z1) and has direction cosines l,m,n of the line, then the cartesian equation of the line is given by

\(\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}\).