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Sagot :
The parametric equation of the line that passes through the points (5,2,7) and (8,–6,1) is [tex]\vec r= < 5,2,7 > +\lambda < 3,-8,-6 >[/tex]
Also, the point (17,–30,31) does not lie on the line.
for given question,
We need to find the parametric equation of the line that passes through the points (5,2,7) and (8,–6,1)
Let [tex]\vec r_0= < 5,2,7 >[/tex]
and
[tex]\vec v= < 8-5,-6-2,1-7 > \\\\\vec v= < 3,-8,-6 >[/tex]
where vector v is the direction vector.
So, the equation of the line would be,
[tex]\vec r=\vec r_0+\lambda \vec v\\\\\vec r= < 5,2,7 > +\lambda < 3,-8,-6 >[/tex]
Now we need to determine whether the point (17, -30, 31) is on the line.
Let [tex]< 17,-30,31 > = < 5,2,7 > +\lambda < 3,-8,-6 >[/tex]
For first component,
⇒ 17 = 5 + 3λ
⇒ 3λ = 12
⇒ λ = 4
For the second component,
⇒ -30 = 2 - 8λ
⇒ -8λ = -32
⇒ λ = 4
For third component,
⇒ 31 = 7 - 6λ
⇒ -6λ = 24
⇒ λ = -4
since the values of the λ are not consistent, the point (17,–30,31) does not lie on the line.
Therefore, the parametric equation of the line that passes through the points (5,2,7) and (8,–6,1) is [tex]\vec r= < 5,2,7 > +\lambda < 3,-8,-6 >[/tex]
Also, the point (17,–30,31) does not lie on the line.
Learn more about the parametric equation of the line here:
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