Let the line L1 passes through the points (5,1,7) and (6,0,8) and the line L2 passes through the points (3,1,3) and . Find the value of for which the two lines intersect.

Answer 1

To find the intersection point of two lines, we need to find the values of x, y, and z that satisfy the equations of both lines.

Let's start with the equation for line L1. The direction vector of L1 can be found by subtracting the coordinates of the two points on the line:

dL1 = (6-5, 0-1, 8-7) = (1, -1, 1)

Using the point-slope form of a line, the equation for L1 can be written as:

L1: (x, y, z) = (5, 1, 7) + t(1, -1, 1) ... (1)

where t is a parameter.

Similarly, we can find the equation for L2. The direction vector of L2 is:

dL2 = (3- ,1,3) = (- ,0,- )

Using the point-slope form, the equation for L2 can be written as:

L2: (x, y, z) = (3, 1, 3) + s(- , 0, - ) ... (2)

where s is a parameter.

Now we can set the x, y, and z values of L1 and L2 equal to each other and solve for t and s:

5 + t = 3 -
1 - t = 1 + 0
7 + t = 3 -

From the second equation, we get t = 0. Substituting this into the first and third equations, we have:

5 = 3 -
7 = 3 -

Solving these equations, we find that = 0.5.

Therefore, the two lines intersect when = 0.5.