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

Hint:
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 \[\dfrac{{x - {x_1}}}{{{x_2} - {x_1}}} = \dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{z - {z_1}}}{{{z_2} - {z_1}}}\].