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+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.

To retrieve the textbook answer, you can follow these steps:

1. Given the lines:

r = (4,7,-1) + t(4,8,-4)
r = (1,5,4) + u(-1,2,3)

2. Find the cross product of the direction vectors of the lines. The direction vectors are (4,8,-4) and (-1,2,3).

To find the cross product:
v = (4,8,-4) x (-1,2,3)

Use the formula for the cross product:
v = ((8)(3) - (-4)(2), (-4)(-1) - (4)(3), (4)(2) - (8)(-1))
v = (32,-8,16)

So, the cross product is v = (32,-8,16).

3. To find the point of intersection (POI), set the two equations of the lines equal to each other:

4 + 4t = 1 - u
7 + 8t = 5 + 2u
-1 - 4t = 4 + 3u

(1) 3 = -u - 4t
(2) 2 = 2u - 8t
(3) -5 = 3u + 4t

4. Solve the system of equations to find the values of t and u. You can use different methods like substitution or elimination. Let's use substitution:

From equation (1), we can solve for u in terms of t:
u = -3 - 4t

Substitute u in equation (2):
2 = 2(-3 - 4t) - 8t
2 = -6 - 8t - 8t
2 = -6 - 16t
16t = -6 - 2
16t = -8
t = -1/2

Substitute t = -1/2 into equation (1):
3 = -u - 4(-1/2)
3 = -u + 2
u = -5

So, t = -1/2 and u = -5.

5. Verify the result by substituting t and u into equation (3):

LHS of (3) = -5
RHS of (3) = 3(-5) + 4(-1/2) = -15 - 2 = -17

Since the LHS does not equal the RHS, the lines are skew and do not intersect.

The textbook answer is not provided in the given information. It seems like the lines do not intersect, and the POI does not exist.