Question

do these lines intersect?
[x,y,z]=[2,-3,2]+s[3,4,-10]
[x,y,z]=[3,4,-2]+s[-4,5,3]

Answer 1

Do you want me to show you what I did?

Answer 2

because I'm not getting the answer my book has

Answer 3

alright. lets see what you did

Answer 4

So I made the parametric equations for both lines

line 1 : x=2+3s y=-3+4s z=2-10s line 2: x=3-4t y=4+5t z=-2+3t

Answer 5

Then I set the equations equal to each other. The x of line 1 with the x of like two and so on. I got these 3 equations

3s+4t=1 4s-5t=7 -10s-3t=-4

Answer 6

yep

Answer 7

then I used elimination on 3s+4t=1 and 4s-5t=7. I multiplied 3s+4t=1 by 4 and 4s-5t=7 by 3. I got t=-17/31 after solving

Answer 8

I then subbed t=-17/31 into 4s-5t=7 to find s. I got s=33/31

Answer 9

yep

Answer 10

After getting t and s. I subbed them into -10s-3t=-4 to see if the right side is equal to the left side. I got -9=-4

Answer 11

but my book has an answer but since I'm given x, y and z. the the point of intersection should also have an x y and z in it. but it gives (33/31, -17/31). But those are my s and t values. so I'm a little confused

Answer 12

because the question asks if the intersect, and what the points are if they do

Answer 13

do you think they intersect?

Answer 14

no, because the left side doesn't equal the right side. Also they are not parallel. So I think they are skew lines. But i think i'm wrong

Answer 15

alright. i would have done it a little different to you but here is what i did.

Answer 16

don't both about your s and t values as i chose s and t values for the opposite lines

Answer 17

so yea i got

[3t-4s,4t-5s,-10t-3s]=[1,7,-4]

Answer 18

ok. now here is the thing

Answer 19

i'll say it later, let me get it into parametric form.

Answer 20

so yea your right with parametric form

Answer 21

3t+4s=1 4t-5s=7 -10t-3s=-4

Answer 22

for simplicity sake let X represent 3t+4s=1 Y represent 4t-5s=7 and Z yea u get the drift.

Answer 23

Now for these two lines to intersect, then when solving simultaneously with X and Y these solution should also be the same for when you solve for X and Z or Y and Z

You only need to do either 2 solvings. like say X and Y and X and Z

or X and Y and Y and Z

Answer 24

So you have solutions for X and Y which were t=33/31 and s=-17/31

Answer 25

now for X and Z we get solutions t=13/31 and s=-2/31

Answer 26

since we get totally different solutions for t and s in both scenario's then these lines do not intersect.

Answer 27

to check this we can plug these numbers into
[3t-4s,4t-5s,-10t-3s]=[1,7,-4]

and you will see that you won't get [1,7,-4]

Answer 28

got any queries?

Answer 29

so why does my book give me an answer that is the s and t values. Because before in other question it would just say no and that was all

Answer 30

what do you mean by the s and t values?

Answer 31

the answer my book gives is (33/31, -17/31)

Answer 32

the s and t values i get

Answer 33

thats weird. those values only seem to work of X and Y

Answer 34

either we are wrong or the book is wrong.

Answer 35

1) double check that you have the question written correctly...2) if you do then the book is wrong

Answer 36

my book always numbers the equations and they always use the first two, x and y then sub into z to see if the sides are equal

Answer 37

i was taught differently. but same concept still applies.

Answer 38

yea so see if you wrote the question down correctly.

Answer 39

yea, i wrote is right

Answer 40

kk. book is silly,

Answer 41

I want to ask my teacher, but i'm scared to say that the book is wrong and i'm right, since i'm just learning this and i'm not the best person in math

Answer 42

doesn't hurt to go through your method with your teacher.

Answer 43

ok. could to stay around. I might have another question

Answer 44

k

Answer 45

I have another question .

Answer 46

My book asks if line r=r(with arrow on top) [-5,1,-2]+k[1,6,5] intersects with the equation [x,y,z]=[2,3,-1]+s[1,3,4]+[-5,4,7]. I know that instead of [x,y,z] i can put r (with arrow on top) but what is with the notation of the first equation, does it mean anything else

Answer 47

can you draw it for me- just the r arrow thing

Answer 48

sorry for my bad drawing

Answer 49

thats what it has

Answer 50

thats just a notation for a vector line

Answer 51

so don't get overwhelmed by it. doesn't contribute to your calculations

Answer 52

but why does it have r=r with arrow

Answer 53

i just put it r with arrow = blah blah

Answer 54

because r itself could be a point but since its r dash then this represents vector notation.

Answer 55

ok thank you so much for your help

Answer 56

You can write vectors on the site

\[\vec{r}\]
\vec{r}