Question

determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give a counterexample. The functions r1(t) = 3cos(t)i + 3 sin(t)j + (3cos(t) + 3sin(t)) k and r2(t) = ti, (9-t^2)j, t+(9-t^2) are different parametrizations of the same curve

Answer 1

FALSE

Explanation:

No, they are not

Counter-example:

Take t = 4

If r1 and r2 were the same curve, there should exist some value
`\bf\large t^*` for which

`\bf\large 4=3cos(t^*)`

Taking the absolute value on both sides

`\bf\large | 4 |=|3cos(t^*)|\Rightarrow 4=3|cos(t^*)|\leq 3`

since the function cos(x) takes values between -1 and 1.

But 4 is greater than 3, and we have a contradiction.

So, r1 and r2 are not the same curve.