Determine whether the lines L1:x=10+3s,y=2+2s,z=22+6s,and L2:x=-7+4t,y=-12+4t,z=-14+9t are intersect,are skew or are parallel.If they intersect,determine the point of intersection;if not leave the remaining answer blanks empty.
Question
Determine whether the lines L1:x=10+3s,y=2+2s,z=22+6s,and L2:x=-7+4t,y=-12+4t,z=-14+9t are intersect,are skew or are parallel.If they intersect,determine the point of intersection;if not leave the remaining answer blanks empty.
Solution
To determine whether the lines L1 and L2 intersect, are skew, or are parallel, we need to compare their direction vectors and check for any consistent values of s and t that satisfy both line equations.
Step 1: Find the direction vectors of the lines The direction vector of L1 is (3, 2, 6). The direction vector of L2 is (4, 4, 9).
Step 2: Compare the direction vectors If the direction vectors are proportional, then the lines are either parallel or coincident. If they are not proportional, then the lines are skew.
The direction vectors are not proportional, so the lines are not parallel.
Step 3: Check for intersection or skewness To determine if the lines intersect or are skew, we need to compare the equations of the lines.
Comparing the x-coefficients: 10 + 3s = -7 + 4t
Comparing the y-coefficients: 2 + 2s = -12 + 4t
Comparing the z-coefficients: 22 + 6s = -14 + 9t
Step 4: Solve the system of equations We can solve the system of equations to find the values of s and t that satisfy both line equations.
From the first equation, we can isolate s: 3s = -17 + 4t s = (-17 + 4t)/3
Substituting this value of s into the second equation, we get: 2 + 2((-17 + 4t)/3) = -12 + 4t 2 - (34/3) + (8t/3) = -12 + 4t (6/3) - (34/3) + (8t/3) = -12 + 4t (-28/3) + (8t/3) = -12 + 4t (-28 + 8t)/3 = -12 + 4t -28 + 8t = -36 + 12t 4t = 8 t = 2
Substituting this value of t back into the first equation, we get: 10 + 3s = -7 + 4(2) 10 + 3s = -7 + 8 3s = 1 s = 1/3
Step 5: Determine the relationship between the lines Since we found consistent values of s and t that satisfy both line equations, the lines L1 and L2 intersect at the point where s = 1/3 and t = 2.
Step 6: Find the point of intersection To find the point of intersection, we can substitute the values of s and t into either line equation. Let's use the equation for L1:
x = 10 + 3(1/3) = 10 + 1 = 11 y = 2 + 2(1/3) = 2 + 2/3 = 8/3 z = 22 + 6(1/3) = 22 + 2 = 24
Therefore, the lines L1 and L2 intersect at the point (11, 8/3, 24).
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