Show that the lines r ?=(2i ?-3k ? )+?(i ?+2j ?+3k ?) and r ?=(2i ?+6j ?+3k ?)+?(2i ?+3j ?+4k ?) intersect. Also, find their point of intersection. The answer given in the book is:((2,6,3)

Asked by Manoj | 18th May, 2013, 12:01: PM

Expert Answer:

 There is something wrong in either the question you typed or the answer. The point you have mentioned does not lie on the first line. We can also prove as following that these 2 lines don't intersect
 
r1=(2i-3k)+?(i+2j+3k)  and r2=(2i+6j+3k)+?(2i+3j+4k) 
 
Any point on r1 = (2+?, 2?, -3+3?) and any point on r2 = (2+2?, 6+3?, 3+4?)
 
If the 2 lines intersect, then the coordinates of the points need to be same
 
Hence, 2? =  6+3? (y coordinate)
? =  3+3?/2
Also, -3+3? = 3+4? (z - coordinate)
Substituting the value of ? 
-3+3(3+3?/2) = 3+4?
3 = -?/2
? = -6
Hence, ? = -6
 
Also, verifying the same by checking their x coordinates
so, x coordinate of r1 = 2+?  = -4
x coordinate on r2 = 2+2? = -10 which is not same as that of the r1. 
 
Hence, the 2 lines don't intersect. 

Answered by  | 20th May, 2013, 10:02: AM

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