Q .35. - Calculus

Question

 (a) Find a vector parametrization for the line containing the points P$(x_0, y_0,z_0 )$ and Q$(x_1, y_1 ,z_1)$.

(b) Apply a restriction to your parameter from part (a) so that the result parametrizes the segment from P to Q 

Step-by-Step Solution

Answer

Part a)$\text{ The required equation is }r\left(t\right)=\left(\left(x_1-x_0\right)t+x_0,\left(y_1-y_0\right)t+y_0,\left(z_1-z_0\right)t+z_0\right)\text{. }$

Part b)$\text{The answer is }r\left(t\right)=\left(\left(x_1-x_0\right)t+x_0,\left(y_1-y_0\right)t+y_0,\left(z_1-z_0\right)t+z_0\right);0\le t\le 1\text{. }$

Part (a)Step 1:Given information

 the line containing the points P$\left(x_0,y_0,z_0\right)$ and Q$(x_1,y_1,z_1)$

Part (a) Step 2:Calculation

$\text{ The point are }P\left(x_0,y_0,z_0\right)\text{ and }Q\left(x_1,y_1,z_1\right)\text{. }$

$\overrightarrow{PQ}=\left(x_1-x_0,y_1-y_0,z_1-z_0\right)$

$\text{ The formula to find the line }\mathcal{L}\text{ equation is as follows, }$

$\text{ Here }P\left(x_0,y_0,z_0\right)\text{ and }\overrightarrow{PQ}=d=\left(x_1-x_0,y_1-y_0,z_1-z_0\right)\text{ then the equation is, }$

$r\left(t\right)=\left(x_0,y_0,z_0\right)+t\left(x_1-x_0,y_1-y_0,z_1-z_0\right)$

$r\left(t\right)=\left(x_0+\left(x_1-x_0\right)t,y_0+\left(y_1-y_0\right)t,z_0+\left(z_1-z_0\right)t\right)$

The equation is written as follows,

$r\left(t\right)=\left(\left(x_1-x_0\right)t+x_0,\left(y_1-y_0\right)t+y_0,\left(z_1-z_0\right)t+z_0\right)$

The equation of a line $\mathcal{L}$ in the form of vector parametrization is,

$r\left(t\right)=\left(\left(x_1-x_0\right)t+x_0,\left(y_1-y_0\right)t+y_0,\left(z_1-z_0\right)t+z_0\right).$

Part (b)Step 1:Given information

$\text{ the line containing the points }P\left(x_0,y_0,z_0\right)\text{ and }Q\left(x_1,y_1,z_1\right)$

Part (b)Step 2:Calculation

The equation of line$\mathcal{L}$ in the form of vector parametrization is,

$r\left(t\right)=\left(\left(x_1-x_0\right)t+x_0,\left(y_1-y_0\right)t+y_0,\left(z_1-z_0\right)t+z_0\right)$

Here the range is restricted so that the line segment starts and ends at the given points.

The line segment starts at P and ends at Q.

Thus $t$ is from 0 to 1 that is $0\le t\le 1$.

$\text{ Therefore, the answer is }r\left(t\right)=\left(\left(x_1-x_0\right)t+x_0,\left(y_1-y_0\right)t+y_0,\left(z_1-z_0\right)t+z_0\right);0\le t\le 1\text{. }$