do these two lines intersect?

Question

r=(2i+3j+k)+s(i+3j+2k)
and
r=(7i+3j+5k)+t(-i +2j)

My calculations say they dont, but not sure if this is correct..

Thanks

Answer 1

To intersect, we'll need (2+s,3+3s,1+2s) = (7-t,3+2t,5).

That immediately tells us that 1+2s = 5, so s = 2.
So we need (4,9,5) = (7-t,3+2t,5).
So 7 - t = 4 gives t=3.
3+2t = 3+2*3 = 9.

So they do intersect - at s=2, t=3, which gives (4,9,5).

Answer 2

The two lines intersect at, r = 4i + 9j + 5k.

r=(2i+3j+k)+s(i+3j+2k)
r=(7i+3j+5k)+t(-i +2j)

To find the point of intersection,
(2i+3j+k)+p(i+3j+2k) = (7i+3j+5k)+q(-i +2j)
(2+p)i + (3+3p)j + (1+2p)k = (7-q)i + (3+2q)j + 5k

k: 1+2p = 5 => p = 2
j: 3+3(2) = 3+2q => q = 3

Check with i: LHS = 2+2 = 4, RHS = 7-3 = 4

So there is an intersection point when p=2 or q=3.
Intersection point
= (2i+3j+k)+p(i+3j+2k)
= 4i + 9j + 5k

Check with q=3,
(7i+3j+5k)+q(-i +2j)
= 4i + 9j + 5k (same as above)

Answer 3

i will do the first one. 7x-2y=5 7x-3y=4 get rid of one variable to sparkling up for the different. hence, x are both 7s. -7x+2y=-5 7x-3y=4 including like words supplies you -y=-a million or y=a million putting a million for y in both equation supplies you 7x-3=4 7x=7 x=a million (a million,a million) i will rewrite your homework for you. 3x+4y=-6 -3x+2y=6 2x+5y=12 -4x+5y=6 attempt to align like words. It makes it extra uncomplicated... try to be able to do something else. hint: they are really basic.

Answer 4

No, they don't. Your calculations are correct...

Answer 5

No, but they meet in the distance, the rule is all parallel lines meet in the distance.

Answer 6

i see no lines