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 (-1,3,a). Find the value of a for which the two lines intersect.
To find the value of a for which the two lines intersect, we need to find the point of intersection between the two lines L1 and L2.
First, we find the direction vector of line L1 by subtracting the coordinates of the two given points on line L1:
L1_direction = (6, 0, 8) - (5, 1, 7) = (1, -1, 1).
Next, we find the direction vector of line L2 by subtracting the coordinates of the two given points on line L2:
L2_direction = (-1, 3, a) - (3, 1, 3) = (-4, 2, a-3).
Since L1_direction and L2_direction are direction vectors of their respective lines, they must be parallel for the lines to intersect. Therefore, the cross product of L1_direction and L2_direction must be the zero vector.
Taking the cross product of L1_direction and L2_direction:
L1_direction x L2_direction = (1, -1, 1) x (-4, 2, a-3)
= (2(a-3) - 2, -4(a-3) - 4, 1 - (-4))
= (2a - 8 - 2, -4a + 12 - 4, 5)
= (2a - 10, -4a + 8, 5).
Since this cross product should be the zero vector (0, 0, 0) for the lines to intersect, we can set each component of the cross product equal to zero:
2a - 10 = 0,
-4a + 8 = 0,
5 = 0.
From the second equation, we have -4a = -8, which gives us a = 2.
Therefore, the value of a for which the two lines intersect is a = 2.
To find the value of 'a' for which the two lines intersect, we need to determine if the lines are parallel or not.
Step 1: Find the direction vectors of the two lines.
The direction vector of line L1 is obtained by subtracting the coordinates of the two given points:
Direction vector of L1 = (6-5, 0-1, 8-7) = (1, -1, 1)
The direction vector of line L2 is obtained by subtracting the coordinates of the two given points:
Direction vector of L2 = (-1-3, 3-1, a-3) = (-4, 2, a-3)
Step 2: Check if the direction vectors are proportional.
If the two direction vectors are proportional, then the lines are parallel, and they do not intersect. If the two direction vectors are not proportional, then the lines are not parallel, and they intersect.
Since we want the lines to intersect, we need the direction vectors to be non-proportional.
Therefore, equating the components of the two direction vectors, we have:
1/(-4) = -1/2 (equating the x-coordinates)
-1/1 = 1/2 (equating the y-coordinates)
1/1 = (a-3)/1 (equating the z-coordinates)
Simplifying the equations, we have:
-1/4 = -1/2 (multiplying both sides by -4)
-1 = 1/2 (multiplying both sides by -1)
1 = a-3
Therefore, a = 1+3 = 4.
Hence, the value of 'a' for which the two lines intersect is 4.