Find the vector equation 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 find the vector equation and Cartesian equation in symmetric form of the line passing through the given points. We will use the direction ratios formula to find the direction ratios and then using the vector equation formula, we will find the vector equation of the line. We will use the Cartesian equation formula to find the Cartesian equation of the line passing through the two given points. Thus the required answer.

Formula Used:
We will use the following formula:
1.Direction Ratios of a line \[\left( {a,b,c} \right)\] are given by \[\left( {{x_2} - {x_1},{y_2} - {y_1},{z_2} - {z_1}} \right)\]
2.Vector equation of the line passing through the two points is given by the formula \[r = X + \lambda \left( {Y - X} \right)\]
3.The Cartesian equation of line whose vector equation of the form \[r = x + \lambda \left( {y - x} \right)\] is given by \[\dfrac{{x - {x_1}}}{{{a_1}}} = \dfrac{{y - {y_1}}}{{{b_1}}} = \dfrac{{z - {z_1}}}{{{c_1}}}\]

Complete step-by-step answer:
We are given the points \[\left( {2,0, - 3} \right)\] and \[\left( {7,3, - 10} \right)\] . Now, we will find the vector equation and Cartesian equation in symmetric form of the line passing through the given points.
Let \[X\] and \[Y\] be the given two points passing through a line.
Now, we will find the direction ratios of the line using the points.
Direction Ratios of a line \[\left( {a,b,c} \right)\] are given by \[\left( {{x_2} - {x_1},{y_2} - {y_1},{z_2} - {z_1}} \right)\]
\[ \Rightarrow \] Direction ratios proportional to the given points \[\left( {2,0, - 3} \right)\] and \[\left( {7,3, - 10} \right)\]are \[\left( {7 - 2,3 - 0, - 10 - \left( { - 3} \right)} \right)\]
\[ \Rightarrow \] Direction ratios proportional to the given points \[\left( {2,0, - 3} \right)\] and \[\left( {7,3, - 10} \right)\]are \[\left( {5,3, - 7} \right)\]
So, the direction ratios of the given points are proportional to \[\left( {5,3, - 7} \right)\].
Now, we will find the vector equation of the line passing through the points \[\left( {2,0, - 3} \right)\] and \[\left( {7,3, - 10} \right)\] .
Vector equation of the line passing through the two points is given by the formula \[r = X + \lambda \left( {Y - X} \right)\]
By using the formula, we get
Vector equation of the line passing through the two points \[\left( {2,0, - 3} \right)\] and \[\left( {7,3, - 10} \right)\]
\[ \Rightarrow r = \left( {2\vec i + 0\vec j - 3\vec k} \right) + \lambda \left( {\left( {7\vec i + 3\vec j - 10\vec k} \right) - \left( {2\vec i + 0\vec j - 3\vec k} \right)} \right)\]
\[ \Rightarrow r = \left( {2\vec i + 0\vec j - 3\vec k} \right) + \lambda \left( {\left( {\left( {7 - 2} \right)\vec i + \left( {3 - 0} \right)\vec j + \left( { - 10 + 3} \right)\vec k} \right)} \right)\]
\[ \Rightarrow r = \left( {2\vec i + 0\vec j - 3\vec k} \right) + \lambda \left( {5\vec i + 3\vec j - 7\vec k} \right)\]
The Cartesian equation of line whose vector equation of the form \[r = x + \lambda \left( {y - x} \right)\] is given by \[\dfrac{{x - {x_1}}}{{{a_1}}} = \dfrac{{y - {y_1}}}{{{b_1}}} = \dfrac{{z - {z_1}}}{{{c_1}}}\]
By using the formula, we get
The Cartesian equation of line whose vector equation of the form \[r = \left( {2\vec i + 0\vec j - 3\vec k} \right) + \lambda \left( {5\vec i + 3\vec j - 7\vec k} \right)\]
\[ \Rightarrow \dfrac{{x - 2}}{5} = \dfrac{{y - 0}}{3} = \dfrac{{z - \left( { - 3} \right)}}{{ - 7}}\]
\[ \Rightarrow \dfrac{{x - 2}}{5} = \dfrac{y}{3} = \dfrac{{z + 3}}{{ - 7}}\]
Therefore, the Vector equation of the line passing through the two points \[\left( {2,0, - 3} \right)\] and \[\left( {7,3, - 10} \right)\]is \[r = \left( {2\vec i + 0\vec j - 3\vec k} \right) + \lambda \left( {5\vec i + 3\vec j - 7\vec k} \right)\]and the Cartesian equation in symmetric form of the line passing through the points \[\left( {2,0, - 3} \right)\] and \[\left( {7,3, - 10} \right)\] is \[\dfrac{{x - 2}}{5} = \dfrac{y}{3} = \dfrac{{z + 3}}{{ - 7}}\].

Note: We know that a vector is an object which has both a magnitude and a direction. The vector equation of a line is used to identify the position vector of every point along the line. Vector equation can be uniquely determined if it passes through a particular point in a specific direction or if it passes through two points. The Cartesian equation is an equation which is represented in three dimensional coordinates.