Question #288557

Find the intersection of the line and plane:

3y−x+z=−4,r(t)=⟨−1,2,1⟩+t⟨−2,−3,1⟩

3y−x+z=−4,r(t)=⟨−1,2,1⟩+t⟨−2,−3,1⟩

P=(

P=(  , , )


1
Expert's answer
2022-01-19T16:26:21-0500

r(t)r(t)

=1,2,1+t2,3,1=⟨-1,2,1⟩+t⟨-2,-3,1⟩

=1,2,1+2t,3t,t=⟨-1,2,1⟩+⟨-2t,-3t,t⟩

Adding, this gives

=12t,23t,1+t=⟨-1-2t,2-3t,1+t⟩

    x=12t\implies x=-1-2t

y=23ty=2-3t

z=1+tz=1+t


Substituting x,y,zx,y,z into 3yx+z3y-x+z, we have

3(23t)(12t)+(1+t)=43(2-3t)-(-1-2t)+(1+t)=-4

    69t+1+2t+1+t=4\implies6-9t+1+2t+1+t=-4

    9t+2t+t+6+1+1=4\implies-9t+2t+t+6+1+1=-4

    6t+8=4\implies-6t+8=-4

    6t=48\implies-6t=-4-8

    6t=12\implies-6t=-12

Dividing both sides by 6-6 .This gives

t=126=2t=\frac{-12}{-6}=2


Substituting tt into 12t,23t,1+t⟨-1-2t,2-3t,1+t⟩

We have

12(2),23(2),1+2⟨−1−2(2),2−3(2),1+2⟩

P=3,4,3P=⟨3,-4,3⟩



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