Question

Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection.

Answer 1

Step 1:

Parametric equations of the lines are

and .

Standard form of parametric equations of the line are , where vector is parallel line to the line.

Compare with standard form.

Parallel line corresponding to the line is .

Consider .

Similarly parallel line corresponding to the line is .

Consider .

If these two parallel lines are parallel, then the lines and also parallel.

Find the cross product of and .

Since the cross product is not equal to zero, then the lines are not parallel.

Answer 2

Continued:

Step 2:

Check for intersection of the  lines:

For point of intersection of and , find the point by solving the lines.

Equate the corresponding parametric equations.

    Equation(1)

       Equation(2)

         Equation(3)

Solve equation(1) and equation(3) and find the values of and .

Multiply the equation(1) by 3 and equation(3) by 2.

Subtract the above equations.

Substitute in equation(1).

substitute and in equation(2).

Thus, the values of do not satisfy the equation(2).

Hence they are not intersecting lines.

The lines and are not intersection lines, they are skew lines.

Solution:

The lines and are skew lines.

Answer 3

Continued:

Step 2:

Check for intersection of the  lines:

For point of intersection of and , find the point by solving the lines.

Equate the corresponding parametric equations.

    Equation(1)

       Equation(2)

         Equation(3)

Solve equation(1) and equation(3) and find the values of and .

Multiply the equation(1) by 3 and equation(3) by 2.

Subtract the above equations.

Substitute in equation(1).

substitute and in equation(2).

Thus, the values of do not satisfy the equation(2).

Hence they are not intersecting lines.

The lines and are not intersection lines, they are skew lines.

Solution:

The lines and are skew lines.