Question

find a parameterization, $mathbf{r}(t)=langle x(t),y(t),z(t)
angle, aleq tleq b$, for the line segment from $(-1,0,-3)$ to $(1,4,2)$. $mathbf{r}(t)=$, where $leq tleq$ (note: you need to enter answers in all three answer blanks to receive feedback.)

Explanation:

Step1: Recall line - segment parameterization formula

The formula for the parameterization of the line - segment from the point $\mathbf{P}(x_1,y_1,z_1)$ to $\mathbf{Q}(x_2,y_2,z_2)$ is $\mathbf{r}(t)=(1 - t)\mathbf{P}+t\mathbf{Q}$, where $0\leq t\leq1$. Here, $x_1=-1,y_1 = 0,z_1=-3$ and $x_2 = 1,y_2 = 4,z_2 = 2$.

Step2: Calculate $x(t)$

$x(t)=(1 - t)x_1+tx_2=(1 - t)(-1)+t(1)=-1 + t+t=-1 + 2t$.

Step3: Calculate $y(t)$

$y(t)=(1 - t)y_1+ty_2=(1 - t)(0)+t(4)=4t$.

Step4: Calculate $z(t)$

$z(t)=(1 - t)z_1+tz_2=(1 - t)(-3)+t(2)=-3 + 3t+2t=-3 + 5t$.

Answer:

$\mathbf{r}(t)=\langle-1 + 2t,4t,-3 + 5t
angle$, where $0\leq t\leq1$