Question

How do you find a vector parametric equation for the line through the point ${\text{P}} = {\text{ }}\left( {0,{\text{ }}4, - 3} \right)$ and ${\text{Q}} = {\text{ }}\left( { - 5,{\text{ }}4,1} \right).$

Answer 1

First we need to convert the given points in vector form. Then the vectors we have to substitute the vector values in the formula, for finding the vector parametric equation and doing some simplification we get the required answer.

Formula used: $\overrightarrow {\text{r}} {\text{ = }}\overrightarrow {\text{A}} {{ + \lambda }}\overrightarrow {\text{b}}$ , where $\overrightarrow {\text{b}} = \overrightarrow {\text{B}} - \overrightarrow {\text{A}}$
$\overrightarrow {\text{r}} {\text{ = }}\overrightarrow {\text{A}} {{ + \lambda }}\left( {\overrightarrow {\text{B}} - \overrightarrow {\text{A}} } \right)$

Complete step-by-step solution:
It is given that the question stated as the point are ${\text{P}} = {\text{ }}\left( {0,{\text{ }}4, - 3} \right)$ and ${\text{Q}} = {\text{ }}\left( { - 5,{\text{ }}4,1} \right).$
Here, representing the given points in vector form we get,
{\text{P}} = {\text{ }}\left( {0,{\text{ }}4, - 3} \right)$$$$ \Rightarrow \overrightarrow {\text{A}} = 0\hat i + 4\hat j - 3\hat k
{\text{Q}} = {\text{ }}\left( { - 5,{\text{ }}4,1} \right)$$$$ \Rightarrow \overrightarrow {\text{B}} = - 5\hat i + 4\hat j + 1\hat k
To find the vector parametric equation of the two given points ${\text{P}} = {\text{ }}\left( {0,{\text{ }}4, - 3} \right)$ and ${\text{Q}} = \left( { - 5,{\text{ }}4,1} \right)$, the formula required is $\overrightarrow {\text{r}} {\text{ = }}\overrightarrow {\text{A}} {{ + \lambda }}\left( {\overrightarrow {\text{B}} - \overrightarrow {\text{A}} } \right)$.
Here $\overrightarrow {\text{r}}$ is the required parametric equation of the two given points ${\text{P}} = {\text{ }}\left( {0,{\text{ }}4, - 3} \right)$ and${\text{Q}} = \left( { - 5,{\text{ }}4,1} \right)$.
So, $\overrightarrow {\text{r}} {\text{ = }}\overrightarrow {\text{A}} {{ + \lambda }}\left( {\overrightarrow {\text{B}} - \overrightarrow {\text{A}} } \right)$ expanding the equation with the given two points P and Q taking Point P as $\overrightarrow {\text{A}}$ and point Q as $\overrightarrow {\text{B}}$.
Putting the value and we can write it as,
$\overrightarrow {\text{r}} {\text{ = }}\left( {0\hat i + 4\hat j - 3\hat k} \right){{ + \lambda }}\left( {\left( { - 5\hat i + 4\hat j + 1\hat k} \right) - \left( {0\hat i + 4\hat j - 3\hat k} \right)} \right)$
Now subtracting the $\overrightarrow {\text{A}}$ from the $\overrightarrow {\text{B}}$ in the second term,
$\overrightarrow {\text{r}} {\text{ = }}\left( {0\hat i + 4\hat j - 3\hat k} \right){{ + \lambda }}\left( {\left( { - 5\hat i + (4 - 4)\hat j + (1 + 3)\hat k} \right)} \right)$
On simplification we get
$\overrightarrow {\text{r}} {\text{ = }}\left( {0\hat i + 4\hat j - 3\hat k} \right){{ + \lambda }}\left( { - 5\hat i + 0\hat j + 4\hat k} \right)$
By adding the both vectors, it can be also written as,
$\overrightarrow {\text{r}} {\text{ = }}\left( { - 5{{\lambda }}\hat i + 4{{\lambda }}\hat j + {{\lambda }}\hat k} \right){\text{ }}$

Hence the required vector parametric equation for the line through the point ${\text{P}} = {\text{ }}\left( {0,{\text{ }}4, - 3} \right)$ and ${\text{Q}} = {\text{ }}\left( { - 5,{\text{ }}4,1} \right)$ are $\overrightarrow {\text{r}} {\text{ = }}\left( { - 5{{\lambda }}\hat i + 4{{\lambda }}\hat j + {{\lambda }}\hat k} \right){\text{ }}$

Note: In this question, we may $\overrightarrow {\text{r}} {\text{ = }}\overrightarrow {\text{A}} {{ + \lambda }}\left( {\overrightarrow {\text{B}} - \overrightarrow {\text{A}} } \right)$ is used to find the equation of line passing through $2$ points with positive vector $\overrightarrow {\text{A}} {\text{ and }}\overrightarrow {\text{B}}$.
So whenever the question highlights on finding the vector equation of line passing through given points, you can use this formula without any second thought.
Also, ${{\lambda }}$ is used here because one vector can be written as ${{\lambda }}$ times the other if it's parallel to each other.
Students will often mistake $\overrightarrow {\text{b}} {\text{ for }}\overrightarrow {\text{B}}$.
But $\overrightarrow {\text{b}}$ is different from$\overrightarrow {\text{B}}$, because $\overrightarrow {\text{b}}$ is the distance between the given two vectors i.e., $\overrightarrow {\text{A}} {\text{ and }}\overrightarrow {\text{B}}$
Therefore, $\overrightarrow {\text{b}} \ne \overrightarrow {\text{B}}$but $\overrightarrow {\text{b}} = \overrightarrow {\text{B}} - \overrightarrow {\text{A}}$.