Math: Equations of Lines
posted by Anonymous on .
I DON'T KNOW HOW TO RETRIEVE THE TEXTBOOK ANSWER!
QUESTION:
Show that the lines:
r = (4,7,1) + t(4,8,4)
r = (1,5,4) + u(1,2,3)
intersect at right angles and find POI.
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ATTEMPT:
v = (4,8,4) x (1,2,3)
v = (32,8,16)
v = (4,1,2)
Finding POI:
l_1:
x = 4 + 4t
y = 7 + 8t
z = 1  4t
l_2:
x = 1  u
y = 5 + 2u
z = 4 + 3u
(1) 3 = u  4t
(2) 2 = 2u  8t
(3) 5 = 3u + 4t
Find "t", sub (1) into (2):
t = 1/2
Find "s", sub "t" into (2):
u = 3
Verify (3):
LS:
= 5
RS:
= 3(3) + 4(1/2)
= 11
Therefore, LS does not equal RS, so skew
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TEXTBOOK ANSWER:
(2,3,1)

first of all if two lines are perpendicular, then the "dot product" of their direction vectors must equal zero, so
(4,8,4)∙(1,2,3) = 4+1612 = 0
so they are perpendicular.
btw, I had t = +1/2 and u = 1 but it did not check out in the third.
so they are skew but at right angles.