Get the complete, step-by-step math solution for: "If the equation of the line passing through the point (0, - {1}{2}, 0) and perpendicular to the lines {r} = ( {i} + a {j} + b {k}) and {r} = ( {i} - {...". Powered by SolveForX AI math tutor.
Step 3: Compare direction vectors to find a and b
The direction vector of the required line is proportional to the direction vector given in the equation −2x−1=dy+4=−4z−c, which is (−2,d,−4). We can equate the ratios of corresponding components. From the first and third components, we have −25a−ab=−4a+ab.
Step 5: Determine d and c
With b=3, the direction vector components are 2a, −(5+32)=−14, and 4a. So the direction vector is (2a, -14, 4a). Comparing this with (−2,d,−4), we have −22a=d−14=−44a. From −22a=−44a, we get −a=−a, which is consistent. From −22a=d−14, we have −a=d−14, so ad=14. Also, from −44a=d−14, we get −a=d−14, which is the same. Since the direction vector components are (−2,d,−4), we can set 2a=−2, which implies a=−1. Substituting a=−1 into ad=14, we get (−1)d=14, so d=−14. The line passes through (0,−21,0) and its equation is −2x−1=dy+4=−4z−c. Since (0,−21,0) lies on the line, we substitute these coordinates into the equation: −20−1=d−21+4=−40−c. This gives 21=d27=−4−c. From 21=d27, we get d=7. This contradicts d=−14. Let's re-evaluate. The direction vector of the required line is proportional to (−2,d,−4). So, (2a,−14,4a)=k(−2,d,−4) for some scalar k. Comparing the first components: 2a=−2k⟹a=−k. Comparing the third components: 4a=−4k⟹a=−k. This is consistent. Comparing the second components: −14=kd. Since a=−k, we have k=−a. So, −14=(−a)d⟹ad=14. We also know that the point (0,−21,0) lies on the line −2x−1=dy+4=−4z−c. Substituting the point: −20−1=d−21+4=−40−c. This simplifies to 21=d27=−4−c. From 21=d27, we get d=7. Now we have ad=14 and d=7, so a(7)=14⟹a=2. From 21=−4−c, we get 21=4c, so c=2.