that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3,5,–1) and (4, 3, –1).

The direction ratios of a line can be found by subtracting the corresponding coordinates of two points through which the line passes i.e. (subtract x coordinates, subtract y coordinates, subtract z coordinates), this is the direction ratio of the line. There can be no direction ratio of a line passing through only one point, there should be at least two points.

The direction ratios of a line joining the origin to the point (2,1,1) are (2–0,1–0,1–0) = (2,1,1)

The direction ratios of a line joining the points (3,5,–1) and (4,3,–1) are (4–3,3–5,–1–{–1}) = (1,–2,0)

By using the dot product we can find the angle between the two lines,

cos θ =

cos θ = 0

θ =

Therefore, the lines are mutually perpendicular.