Question

use the result of Exercise 35 to find parametric equations for the line segment connecting point P to point Q.

$P(0,0,0),Q(1,2,3)$

Short answer

$\text{The answer is}x\left(t\right)=t,y\left(t\right)=2t,z\left(t\right)=3t,0\le t\le 1\text{.}$

Step-by-step solution

Step 1:Given information

$\text{The points}P(0,0,0)\text{and}Q(1,2,3)\text{.}$

Step 2:Calculation

$\text{The point are}P(0,0,0)\text{and}Q(1,2,3)\text{.}$

$\overrightarrow{PQ}=(1-0,2-0,3-0)$

$\overrightarrow{PQ}=(1,2,3)$

The formula to find the line $\mathcal{L}$equation is as follows, $r\left(t\right)=P_0+td$Where, $P_0$is the point and $d$is the direction vector.

$\text{Here}P(0,0,0)\text{and}\overrightarrow{PQ}=d=(1,2,3)\text{then the equation is,}$

$r\left(t\right)=(0,0,0)+t(1,2,3)$

$r\left(t\right)=(0+t,0+2t,0+3t)$

$\text{The equation is}r\left(t\right)=(t,2t,3t)\text{.}$

$\text{The vector function}r\left(t\right)\text{in three -dimensional plane represents}r\left(t\right)=\left(x\right(t),y(t),z(t\left)\right)\text{.}$

$\text{Then,}r\left(t\right)=\left(x\right(t),y(t),z(t\left)\right)=(t,2t,3t)$

$\text{Thus the parametric equations are}x\left(t\right)=t,y\left(t\right)=2t,z\left(t\right)=3t\text{.}$

The restriction for the parameter $t$so that the result parametrizes the segment P to point Q is given from 0 to 1 that is from $0\le t\le 1$.

Thus, the parametric equations are $x\left(t\right)=t,y\left(t\right)=2t,z\left(t\right)=3t$where $0\le t\le 1$

$\text{Therefore, the answer is}x\left(t\right)=t,y\left(t\right)=2t,z\left(t\right)=3t,0\le t\le 1\text{.}$

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