Three Dimensional Geometry

The relationship between the direction cosines of a line is given by: Using the identity for double angles: We can express the sum as follows: Thus, the value of is .

The given lines are: The point on is closest to . To find the point on closest to , we use the property that the vector connecting the closest points on the lines is perpendicular to both direction vectors.Direction Vectors: for for Parametric Points:A point on : A point on : Vector Between Points: Perpendicular Conditions: perpendicular to : Expanding this: Simplifies to: Therefore: perpendicular to : Expanding this: Simplifies to: Solving Equations:From Equation 2: Substitute into Equation 1: Simplifies to: Substituting into : Closest Point on :Substitute into the equation for : Thus, the point on that is closest to is .