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Determine if the lines are parallel, skew or intersect. If they intersect, find the point of intersection

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x=−7+3t

y=−1+3t

z=2−3t

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x=−1+4s

y=6+4s

z=2−s

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Answer :

We have determined that the lines intersect and the point of intersection is (-7/7, 24/7, 17/7).

The given equations represent two lines in three-dimensional space. To determine if the lines are parallel, skew, or intersect, we need to analyze their direction vectors.

For the first line:
x = -7 + 3t
y = -1 + 3t
z = 2 - 3t

The direction vector of this line is <3, 3, -3>.

For the second line:
x = -1 + 4s
y = 6 + 4s
z = 2 - s

The direction vector of this line is <4, 4, -1>.

To determine if the lines are parallel, we can compare the direction vectors. If the direction vectors are scalar multiples of each other, the lines are parallel. If not, they are not parallel.

In this case, the direction vectors <3, 3, -3> and <4, 4, -1> are not scalar multiples of each other. Therefore, the lines are not parallel.

To determine if the lines intersect, we can set the equations for x, y, and z equal to each other and solve for t and s. If there is a solution, the lines intersect. If not, they do not intersect.

Setting the x, y, and z equations equal to each other, we get:

-7 + 3t = -1 + 4s
-1 + 3t = 6 + 4s
2 - 3t = 2 - s

Simplifying these equations, we get:

3t - 4s = -6
3t - 4s = 7
-3t + s = 0

Solving this system of equations, we find that

t = -3/7 and

s = -9/7.

Therefore, the lines intersect at the point (-7/7, 24/7, 17/7).

In conclusion, the lines are not parallel, and they intersect at the point (-1, 6, 2).

Learn more about the point of intersection from the given link-

https://brainly.com/question/29185601

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