Q .36. - Calculus

Question

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

$P(0,2,3),Q(4,5,-1)$

Step-by-Step Solution

Answer

$\text{ The answer is }x\left(t\right)=4t,y\left(t\right)=2+3t,z\left(t\right)=3-4t,0\le t\le 1\text{. }$

Step 1:Given information

$\text{ The points }P(0,2,3)\text{ and }Q(4,5,-1)\text{. }$

Step 2:Calculation

$\text{ The point are }P(0,2,3)\text{ and }Q(4,5,-1)$

$\overrightarrow{PQ}=(4-0,5-2,-1-3)$

$\overrightarrow{PQ}=(4,3,-4)$

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 dis the direction vector.

$\text{ Here }P(0,2,3)\text{ and }\overrightarrow{PQ}=d=(4,3,-4)\text{ then the equation is, }$

$r\left(t\right)=(0,2,3)+t(4,3,-4)$

$r\left(t\right)=(0+4 t, 2+3 t, 3-4 t)$

The equation is written as follows,

$r\left(t\right)=(4 t, 2+3 t, 3-4 t)$

The vector function $r\left(t\right)$ in three -dimensional plane represents $r\left(t\right)=\left(x\right(t), y(t), z(t\left)\right)$. Then, $r\left(t\right)=\left(x\right(t), y(t), z(t\left)\right)=(4 t, 2+3 t, 3-4 t$)

Thus the parametric equations are $x\left(t\right)=4 t, y\left(t\right)=2+3 t, z\left(t\right)=3-4 t$.

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)=4 t, y\left(t\right)=2+3 t, z\left(t\right)=3-4 t$$ where $0\le t\le 1$.

$\text{ Therefore, the answer is }x\left(t\right)=4t,y\left(t\right)=2+3t,z\left(t\right)=3-4t,0\le t\le 1\text{. }$