Question

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)

Answer 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+16-12 = 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.