Question

Find the vector and Cartesian equation in symmetric form of the line passing through the points $\left( {2,0, - 3} \right)$ and $\left( {7,3, - 10} \right)$.

Answer 1

Here, we will first calculate the direction ratios of the given line. Then, we will use the formula for vector equation and cartesian equation of a line, and substitute the values to find the vector and Cartesian equation in symmetric form of the line passing through the points $\left( {2,0, - 3} \right)$ and $\left( {7,3, - 10} \right)$.

Formula Used:
We will use the following formulas:
1. The direction ratios of a line segment joining the points $\left( {{x_1},{y_1},{z_1}} \right)$ and $\left( {{x_2},{y_2},{z_2}} \right)$ are given by ${x_2} - {x_1}$, ${y_2} - {y_1}$, and ${z_2} - {z_1}$.
2. The vector equation of a line joining the points $\left( {{x_1},{y_1},{z_1}} \right)$ and $\left( {{x_2},{y_2},{z_2}} \right)$ is given by $\overrightarrow r = \overrightarrow a + \lambda \overrightarrow b$, where $\overrightarrow a = {x_1}\hat i + {y_1}\hat j + {z_1}\hat k$, and $\overrightarrow b = a\hat i + b\hat j + c\hat k$.
3. The Cartesian equation of a line joining the points $\left( {{x_1},{y_1},{z_1}} \right)$ and $\left( {{x_2},{y_2},{z_2}} \right)$ is given by $\dfrac{{x - {x_1}}}{a} = \dfrac{{y - {y_1}}}{b} = \dfrac{{z - {z_1}}}{c}$, where $a$, $b$, and $c$ are the direction ratios.

Complete step by step solution:
First, we will find the direction ratios of the line passing through the points $\left( {2,0, - 3} \right)$ and $\left( {7,3, - 10} \right)$.
Let the direction ratios of the line passing through the points $\left( {2,0, - 3} \right)$ and $\left( {7,3, - 10} \right)$ be $a$, $b$, and $c$ respectively.
The direction ratios of a line segment joining the points $\left( {{x_1},{y_1},{z_1}} \right)$ and $\left( {{x_2},{y_2},{z_2}} \right)$ are given by ${x_2} - {x_1}$, ${y_2} - {y_1}$, and ${z_2} - {z_1}$.
Substituting ${x_1} = 2$, ${y_1} = 0$, ${z_1} = - 3$, ${x_2} = 7$, ${y_2} = 3$, and ${z_2} = - 10$ in the formula for direction ratios, we get
${x_2} - {x_1} = 7 - 2 = 5$
${y_2} - {y_1} = 3 - 0 = 3$
${z_2} - {z_1} - 10 - \left( { - 3} \right) = - 10 + 3 = - 7$
Therefore, we get
$a = 5$
$b = 3$
$c = - 7$
Now, we will find the vector equation of the line passing through the points $\left( {2,0, - 3} \right)$ and $\left( {7,3, - 10} \right)$.
The vector equation of a line joining the points $\left( {{x_1},{y_1},{z_1}} \right)$ and $\left( {{x_2},{y_2},{z_2}} \right)$ is given by $\overrightarrow r = \overrightarrow a + \lambda \overrightarrow b$, where $\overrightarrow a = {x_1}\hat i + {y_1}\hat j + {z_1}\hat k$, and $\overrightarrow b = a\hat i + b\hat j + c\hat k$.
Substituting ${x_1} = 2$, ${y_1} = 0$, ${z_1} = - 3$, $a = 5$, $b = 3$, and $c = - 7$ in the equations $\overrightarrow a = {x_1}\hat i + {y_1}\hat j + {z_1}\hat k$ and $\overrightarrow b = a\hat i + b\hat j + c\hat k$, we get
$\overrightarrow a = 2\hat i + 0\hat j - 3\hat k = 2\hat i - 3\hat k$
$\overrightarrow b = 5\hat i + 3\hat j - 7\hat k$
Substituting $\overrightarrow a = 2\hat i - 3\hat k$ and $\overrightarrow b = 5\hat i + 3\hat j - 7\hat k$ in the vector equation of a line joining two points, we get
$\overrightarrow r = \left( {2\hat i - 3\hat k} \right) + \lambda \left( {5\hat i + 3\hat j - 7\hat k} \right)$
Multiplying the terms using the distributive law of multiplication, we get
$\Rightarrow \overrightarrow r = 2\hat i - 3\hat k + 5\lambda \hat i + 3\lambda \hat j - 7\lambda \hat k$
Factoring the terms, we get
$\Rightarrow \overrightarrow r = \left( {2 + 5\lambda } \right)\hat i + \left( {3\lambda } \right)\hat j - \left( {3 + 7\lambda } \right)\hat k$
Thus, the vector equation of the given line is $\overrightarrow r = \left( {2 + 5\lambda } \right)\hat i + \left( {3\lambda } \right)\hat j - \left( {3 + 7\lambda } \right)\hat k$.
Now, we will find the Cartesian equation of the line passing through the points $\left( {2,0, - 3} \right)$ and $\left( {7,3, - 10} \right)$.
The Cartesian equation of a line joining the points $\left( {{x_1},{y_1},{z_1}} \right)$ and $\left( {{x_2},{y_2},{z_2}} \right)$ is given by $\dfrac{{x - {x_1}}}{a} = \dfrac{{y - {y_1}}}{b} = \dfrac{{z - {z_1}}}{c}$, where $a$, $b$, and $c$ are the direction ratios.
Substituting ${x_1} = 2$, ${y_1} = 0$, ${z_1} = - 3$, $a = 5$, $b = 3$, and $c = - 7$ in the Cartesian equation of a line joining two points, we get
$\dfrac{{x - 2}}{5} = \dfrac{{y - 0}}{3} = \dfrac{{z - \left( { - 3} \right)}}{{ - 7}}$
Simplifying the expression, we get
$\Rightarrow \dfrac{{x - 2}}{5} = \dfrac{y}{3} = \dfrac{{z + 3}}{{ - 7}}$
Thus, the Cartesian equation of the given line is $\dfrac{{x - 2}}{5} = \dfrac{y}{3} = \dfrac{{z + 3}}{{ - 7}}$.

Note:
We have used the distributive law of multiplication to multiply $\lambda$ by $\left( {5\hat i + 3\hat j - 7\hat k} \right)$. The distributive law of multiplication states that $a\left( {b + c + d} \right) = a \cdot b + a \cdot c + a \cdot d$.
We can also find the vector equation of a line joining the points $\left( {{x_1},{y_1},{z_1}} \right)$ and $\left( {{x_2},{y_2},{z_2}} \right)$ directly using the formula $\overrightarrow r = \overrightarrow a + \lambda \left( {\overrightarrow b - \overrightarrow a } \right)$, where $\overrightarrow a = {x_1}\hat i + {y_1}\hat j + {z_1}\hat k$, $\overrightarrow b = {x_2}\hat i + {y_2}\hat j + {z_2}\hat k$, and $\lambda \in R$.
Similarly, we can also find the cartesian equation of a line joining the points $\left( {{x_1},{y_1},{z_1}} \right)$ and $\left( {{x_2},{y_2},{z_2}} \right)$ directly using the formula $\dfrac{{x - {x_1}}}{{{x_2} - {x_1}}} = \dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{z - {z_1}}}{{{z_2} - {z_1}}}$.