Chapter 3, Problem 202RE

True or False? Justify your answer with a proof or a counterexample.

202. A parametric equation that passes through points P and Q can be given by $r\left(t\right)=\langle t^2,3t+1,t-2\rangle$ , where $P\left(1,4,-1\right)$ and $Q\left(16,11,2\right)$ .

To determine

Whether the given statement is true or false and justify it with a proof.

Answer to Problem 202RE

The given statement is false because it does not satisfy the point $Q\left(16,11,2\right)$.

Explanation of Solution

Given information:

The parametric equation of line is and the given points are $P\left(1,4,-1\right)$ and $Q\left(16,11,2\right)$.

Calculation:

The given parametric equation;

…… $\left(1\right)$

The given parametric equation passes though points;

$P\left(1,4,-1\right)$ and $Q\left(16,11,2\right)$

As the parametric equation passes through points, then it should satisfy the points.

Check for point $P\left(1,4,-1\right)$:

Substitute $P\left(1,4,-1\right)$ in equation $\left(1\right)$ ;

$\begin{array}{c}{t}^2=1\\ t=1\end{array}$

$\begin{array}{c}3t+1=4\\ 3t=3\\ t=1\end{array}$

$\begin{array}{c}t-2=-1\\ t=1\end{array}$

So, the equation satisfies the point $P\left(1,4,-1\right)$.

Check for point $Q\left(16,11,2\right)$:

Substitute $Q\left(16,11,2\right)$ in equation $\left(1\right)$ ;

$\begin{array}{c}{t}^2=16\\ t=4\end{array}$

$\begin{array}{c}3t+1=11\\ 3t=10\\ t=\frac{10}{3}\end{array}$

$\begin{array}{l}t-2=2\\ t=4\end{array}$

So, the equation does not satisfy the point $Q\left(16,11,2\right)$.

The given statement is false.