Question

How do you find the parametric equations of the line that passes through the points $(1,3,2)$ and $( - 4,3,0)$?

Answer 1

Hint:
According to the question we have to determine the parametric equations of the line that passes through the points $(1,3,2)$ and $( - 4,3,0)$. So, first of all to determine the parametric equations of the line that passes through the points we have to find the required vector v which is passing through the points $({a_1},{b_1},{c_1})$by using the formula which is as mentioned below:
Formula used:
$ \Rightarrow v = ({a_1} - {a_0},{b_1} - {b_0},{c_1} - {c_0}).................(A)$
Where, v is the vector and $({a_1},{b_1},{c_1})$, $({a_0},{b_0},{c_0})$are the points.
Now, we have to determine the parameters which is done with the help of parameters t by using the equation below:
Formula used:
$ \Rightarrow x = {a_0} + vt.............(B)$
Now same as we have to determine the other parameter with the help of parameters by using the formula as mentioned below:
Formula used:
$ \Rightarrow y = {b_0} + vt.............(C)$
Now same as we have to determine the other parameter with the help of parameters by using the formula as mentioned below:
Formula used:
$ \Rightarrow z = {c_0} + vt.............(D)$

Complete step by step solution:
Step 1: First of all to determine the parametric equations of the line that passes through the points we have to find the required vector v which is passing through the points $({a_1},{b_1},{c_1})$by using the formula (A) which is as mentioned in the solution hint. Hence,
$
   \Rightarrow v = (3 - 3,2 - 1, - 6 + 1) \\
   \Rightarrow v = (0,1, - 5) \\
 $
Step 2: Now, we have to determine the parameters which is done with the help of parameters t by using equation (B) in the solution hint. Hence,
$
   \Rightarrow x = 3 + t(0) \\
   \Rightarrow x = 3 \\
 $
Step 3: Now same as we have to determine the other parameter with the help of parameters by using the formula (C) as mentioned in the solution hint. Hence,
$
   \Rightarrow y = 1 + t(1) \\
   \Rightarrow y = 1 + t \\
 $
Step 4: Now same as we have to determine the other parameter with the help of parameters by using the formula (D) as mentioned in the solution hint. Hence,
$
   \Rightarrow z = - 1 + t( - 5) \\
   \Rightarrow y = - 1 - 5t \\
 $

Hence, with the help of the formulas (A), (B), (C), and (D) we have determined the required parametric equations of the line that passes through the points $(1,3,2)$ and $( - 4,3,0)$ are $x = 3,y = 1 + t,z = - 1 - 5t$.

Note:
1) To determine the required parametric equations of the line that passes through the given points it is necessary that we have to determine the value of the vector with the help of the formula (A) as mentioned in the solution hint.
2) Parameterization is done with the help of the parameter t by using the equations $x = {a_0} + vt,y = {b_0} + vt,z = {c_0} + vt$.