Q .2. - Calculus

Question

2. Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

 (a) A line in ${\mathrm{ℝ}}^3$ that is parallel to the x z-plane. 

(b) A line in ${\mathrm{ℝ}}^3$ through the origin.

(c) A line parallel to the z-axis 

Step-by-Step Solution

Answer

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

Part b)$\text{The required answer is }r\left(t\right)=(at,bt,ct)\text{. }$

Part c)$\text{The required answer is }r\left(t\right)=\left(x_0,y_0,z_0+t\right)\text{. }$

Part (a) Step 1:Given information

A line in ${\mathrm{ℝ}}^3$ that is parallel to the x z-plane 

Part (a) Step 2:Calculation

$\text{ The line equation for the point }\left(x_0,y_0,z_0\right)\text{ and the direction vector }(a,o,c)\text{ is }$

$r\left(t\right)=\left(x_0,y_0,z_0\right)+t(a,o,c)$

$r\left(t\right)=\left(x_0+at,y_0,z_0+ct\right)$

Example:

$\text{ For }P(-1,3,7),d=(2,0,4)\text{ the equation is, }$

$r\left(t\right)=(-1,3,7)+t(2,0,4)$

$r\left(t\right)=(-1,3,7)+t(2,0,4)$

$r\left(t\right)=(-1+2 t, 3,7+4 t)$

$\text{ Therefore, the required equation is }r\left(t\right)=\left(x_0+at,y_0,z_0+ct\right)\text{. }$

Part (b) Step 1:Given information

A line in ${\mathrm{ℝ}}^3$ through the origin. 

Part (b) Step 2:Calculation

$\text{ Consider a point }(0,0,0)\text{ and the direction vector }(a,b,c)\text{. }$

$\text{ The line equation for the point }(0,0,0)\text{ and the direction vector }(a,b,c)\text{ is }$

$r\left(t\right)=(0,0,0)+t(a, b, c)$

$r\left(t\right)=(0+a t, 0+b t, 0+c t)$

$\text{ Therefore, the required answer is }r\left(t\right)=(at,bt,ct)\text{. }$

Part (c) Step 1:Given information

A line parallel to the z-axis 

Part (c) step 2:Calculation

$\text{ Consider a line parallel to }z\text{-axis, then the direction vectors are }(0,0,1)$

$\text{ The line equation for the point }\left(x_0,y_0,z_0\right)\text{ and the direction vector }(0,0,1)\text{ is }$

$r\left(t\right)=\left(x_0,y_0,z_0\right)+t(0,0,1)$

$r\left(t\right)=\left(x_0,y_0,z_0+t\right)$

$\text{ Therefore, the required answer is }r\left(t\right)=\left(x_0,y_0,z_0+t\right)\text{. }$