The distance of the line 2x−2=3y−6=4z−3 from the point (1,4,0) along the line 4x=2y−2=3z+3 is:
Get the complete, step-by-step math solution for: "The distance of the line (x-2)/(2)=(y-6)/(3)=(z-3)/(4) from the point (1{,}4{,}0) along the line (x)/(4)=(y-2)/(2)=(z+3)/(3) is:". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Define the lines and point
We are given two lines, L1 and L2, and a point P0. We need to find the distance from P0 to L1 measured along L2. First, we write the parametric equations for both lines. Let L1 be parameterized by λ and L2 by μ.
Step 2: Find a general point on L2
Any point on line L2 can be represented by the coordinates (4μ,2μ+2,3μ−3) by setting each part of the equation for L2 equal to μ. This point will be used to form a line segment from P0 to L1.
Step 3: Find the equation of the line segment from P0 to L1
The line segment connecting P0(1,4,0) to a general point P2(4μ,2μ+2,3μ−3) on L2 has direction vector P0P2. This vector represents the line along which we are measuring the distance.
Step 4: Determine the intersection point on L1
The line segment P0P2 must intersect L1. This means the point P2 must lie on L1. We substitute the coordinates of P2 into the equation for L1 and solve for μ. We equate the first two ratios to find μ.
Step 5: Calculate the coordinates of P2
Now that we have the value of μ, we can find the exact coordinates of the point P2 on L2 that lies on L1. This point P2 is the point on L1 that is reached from P0 by traveling along L2.
Step 6: Calculate the distance between P0 and P2
Finally, we calculate the distance between the point P0(1,4,0) and the point P2(−27,41,−845) using the distance formula in 3D. This distance is the required distance of L1 from P0 along L2.