Direction Cosines And Direction Ratios Of A Line

Step 1: Find the direction ratios.
The direction ratios of the line are obtained by subtracting the coordinates of the two given points:

Step 2: Find the direction cosines.
The direction cosines are the ratios of the direction ratios to the magnitude of the direction vector. The magnitude of the direction vector is: Thus, the direction cosines are:

Step 3: Show that the line does not pass through the origin.
For the line to pass through the origin, we must check if the point satisfies the equation of the line. Using the parametric form of the line: Setting and solving for : Substitute into the equations for and : Thus, the line does not pass through the origin.

Final Answer: The direction ratios are , and the direction cosines are . The line does not pass through the origin.

Step 1: Use the direction cosines formula.
Let the direction cosines of the line be , , and , where , , and are the angles the line makes with the , , and axes, respectively. From the given information:

Step 2: Apply the direction cosines equation.
The sum of the squares of the direction cosines is always 1: Substitute the known values for and :

Step 3: Find the angle .
Since , we find:

Final Answer: The angle made by the line with the positive direction of the -axis is .