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Anuta_ua [19.1K]
3 years ago
5

Determine if the lines r1(t) = <3, 0, 2> + t <1, 2, −2> and r2(s) = <0, 1, −1> + s <4, 1, 1> intersect,

and if they do, find the point of intersection.
Mathematics
1 answer:
Temka [501]3 years ago
4 0

Answer:

Yes lines are intersecting, point of intersection is <4,2,0>.

Step-by-step explanation:

Given parametric equations of line are:

r_{1}(t) =  + t  \\ r_{1}(t) = \\ r_{1}(t) = ---(1)

r_{2}(s) =  + s \\ r_{2}(s) = \\ r_{2}(s) = ---(2)\\

If lines are intersecting then parametric coordinates of (1) are equal to (2)

3+t=4s---(A)\\2t=1+s---(B)\\2-2t=s-1---(C)\\

Considering A and B to find values of t and s

From A

t=4s-3---(D)

Putting in (B)

2(4s-3)=1+s

8s-6=1+s

7s=7

s=1

Then

t=4-3

t=1

If lines are intersecting then these values of s and t must satisfy (C)

2-2(1)=1-1

0=0

This shows lines are intersecting.

At this value of t, (1) becomes

r_{1}(1) = \\=

Putting s=1 in (2)

r_{2}(1)=4, 1+1,-1+1>\\=

Point of intersection is <4,2,0>.

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