Determine if the two vectors are skew lines

Answered question

2022-04-30

Determine if the two vectors are skew lines or if they intersect each other. !D = h2;3;1i+d h5;3;6i and −!F = h−5;−3;−5i+ f h15;3;−3i

 

Answer & Explanation

alenahelenash

alenahelenash

Expert2023-05-02Added 556 answers

To determine whether the two vectors are skew lines or intersect each other, we need to first find the equations of the lines that contain each of the vectors.
For vector D=(231)+d(536), the equation of the line is:
(xyz)=(231)+d(536)
Simplifying, we get:
x=5d+2
y=3d+3
z=6d+1
For vector F=(535)+f(1533), the equation of the line is:
(xyz)=(535)+f(1533)
Simplifying, we get:
x=15f5
y=3f3
z=3f5
Now we can check whether the two lines intersect or are skew by solving for d and f in terms of each other. If there is no solution, the lines are skew, and if there is a solution, the lines intersect.
Setting the equations for x, y, and z of the two lines equal to each other, we get:
5d+2=15f5
3d+3=3f3
6d+1=3f5
Solving for d in terms of f using the second equation, we get:
d=f2
Substituting this expression into the first and third equations, we get:
5(f2)+2=15f5
6(f2)+1=3f5
Simplifying, we get:
10f=27
f=2710
Substituting this value of f into the equation for d, we get:
d=27102=710
Therefore, the two lines intersect at the point (372242312).

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