Let L1:2x−1=3y−2=4z−3 and L2:3x−2=4y−4=6z−5 be two lines. Then which of the following points lies on the line of the shortest distance between L1 and L2?
Get the complete, step-by-step math solution for: "Let L_1: (x-1)/(2) = (y-2)/(3) = (z-3)/(4) and L_2: (x-2)/(3) = (y-4)/(4) = (z-5)/(6) be two lines. Then which of the following points lies on the lin...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify points and direction vectors
First, we identify a point on each line and their respective direction vectors. For line L1, a point is A1(1,2,3) and its direction vector is b1=(2,3,4). For line L2, a point is A2(2,4,5) and its direction vector is b2=(3,4,6).
Step 2: Calculate the vector connecting the points
Next, we find the vector connecting a point on L1 to a point on L2. This vector is A1A2=a2−a1.
Step 3: Calculate the cross product of direction vectors
The shortest distance between two skew lines is perpendicular to both direction vectors. We find a vector normal to both b1 and b2 by calculating their cross product, b1×b2.
Step 4: Calculate the shortest distance
We calculate the shortest distance between the lines using the formula for skew lines. Since the shortest distance is 0, the lines are intersecting. The line of shortest distance in this case is simply the point of intersection.
Step 5: Find the point of intersection
To find the point of intersection, we equate the general points on L1 and L2. This gives us a system of three linear equations. Solving the first two equations: 2λ−3μ=1 and 3λ−4μ=2. Multiplying the first by 3 and the second by 2 gives 6λ−9μ=3 and 6λ−8μ=4. Subtracting these yields −μ=−1, so μ=1. Substituting μ=1 into 2λ−3μ=1 gives 2λ−3=1, so 2λ=4, and λ=2. We verify with the third equation: 3+4(2)=11 and 5+6(1)=11. Both sides are equal, so the values are consistent.
Step 6: Determine the intersection point
Substitute λ=2 into the equation for L1 to find the coordinates of the intersection point. Alternatively, substituting μ=1 into the equation for L2 would yield the same point. The point of intersection is (5,8,11). This point lies on the line of shortest distance (which is just the point itself since the lines intersect).