which of the following points are onthe trajectory : P(1,4,2), Q(9,-8, 28), R(4,

Question

which of the following points are onthe trajectory : P(1,4,2), Q(9,-8, 28), R(4,-5,9)? How do you know?

Explanation / Answer

We are given the curve r(t) = t^2i + (1-3t)j +(1+t^3)k . This curve means : x = t^2 , y = (1-3t) , and z = (1 + t^3) . The want to know which of the points P = (1,4,2) , Q = (9,-8,28) , and R = (4,-5 ,9) lies on the curve . Start with point P = (1,4,2) : x=1=t^2 ---> t = + or - 1 . Substititute t = +1 into y = 1-3(1)=1-3=-2 does not equal 4 . Substitute t = -1 into : y= 1-3(-1)= 1 + 3=4 ; but , z = 1+(-1)^3=1-1=0 no good . We thus have P = (1,4,2) does not lie on the curve . Next , try Q = (9,-8,28) : x= 9 = t^2 ---> t = + or -3 . Substitute t = +3 into --- y= 1-3(3)=1-9=-8 and z = 1+3^3 = 1+27=28 . We thus have Q = (9,-8,28) lies on the curve --- solution . Finally , try R = (4,-5 ,9): x=t^2 =4 ----> t = + or - 2 . Substitute t =+2 into y=1-3(2)=1-6=-5 and z =1 +2^3=1+8=9 . We thus also have R = (4,-5 ,9 also lies on the curve --- solution . We therefore have Q = (9,-8,28) and R = (4,-5 ,9) lie on the curve ---- solution .

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