Find the vector and cartesian equations of the line that passes through :
(i) the origin and (5, -2, 3)
(ii) the points (1,2,3) and (2,-1,4)

Answer

To find the vector and Cartesian equations of the line that passes through the given points, we will follow these steps for both parts of the question.

### Part (i): Line through the Origin and (5, -2, 3)

**Step 1: Identify the points.**
- Point A (the origin): \( (0, 0, 0) \)
- Point B: \( (5, -2, 3) \)

**Step 2: Find the direction ratios.**
- The direction ratios can be found by subtracting the coordinates of point A from point B:
\[ \text{Direction ratios} = (5 - 0, -2 - 0, 3 - 0) = (5, -2, 3) \]

**Step 3: Write the vector equation of the line.**
- The vector equation of a line can be expressed as:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \]
where \( \mathbf{a} \) is a position vector of point A and \( \mathbf{b} \) is the direction vector.
- Here, \( \mathbf{a} = 0 \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k} = \mathbf{0} \) and \( \mathbf{b} = 5 \mathbf{i} - 2 \mathbf{j} + 3 \mathbf{k} \).
- Thus, the vector equation is:
\[ \mathbf{r} = \lambda (5 \mathbf{i} - 2 \mathbf{j} + 3 \mathbf{k}) \]

**Step 4: Write the Cartesian equation of the line.**
- The Cartesian form can be derived from the direction ratios:
\[ \frac{x - 0}{5} = \frac{y - 0}{-2} = \frac{z - 0}{3} \]
This simplifies to:
\[ \frac{x}{5} = \frac{y}{-2} = \frac{z}{3} \]

### Part (ii): Line through the points (1, 2, 3) and (2, -1, 4)

**Step 1: Identify the points.**
- Point A: \( (1, 2, 3) \)
- Point B: \( (2, -1, 4) \)

**Step 2: Find the direction ratios.**
- The direction ratios can be found by subtracting the coordinates of point A from point B:
\[ \text{Direction ratios} = (2 - 1, -1 - 2, 4 - 3) = (1, -3, 1) \]

**Step 3: Write the vector equation of the line.**
- Using the same formula as before:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \]
where \( \mathbf{a} = 1 \mathbf{i} + 2 \mathbf{j} + 3 \mathbf{k} \) and \( \mathbf{b} = 1 \mathbf{i} - 3 \mathbf{j} + 1 \mathbf{k} \).
- Thus, the vector equation is:
\[ \mathbf{r} = (1 \mathbf{i} + 2 \mathbf{j} + 3 \mathbf{k}) + \lambda (1 \mathbf{i} - 3 \mathbf{j} + 1 \mathbf{k}) \]

**Step 4: Write the Cartesian equation of the line.**
- The Cartesian form can be derived from the direction ratios:
\[ \frac{x - 1}{1} = \frac{y - 2}{-3} = \frac{z - 3}{1} \]

### Summary of Results:
- For part (i):
- Vector Equation: \( \mathbf{r} = \lambda (5 \mathbf{i} - 2 \mathbf{j} + 3 \mathbf{k}) \)
- Cartesian Equation: \( \frac{x}{5} = \frac{y}{-2} = \frac{z}{3} \)

- For part (ii):
- Vector Equation: \( \mathbf{r} = (1 \mathbf{i} + 2 \mathbf{j} + 3 \mathbf{k}) + \lambda (1 \mathbf{i} - 3 \mathbf{j} + 1 \mathbf{k}) \)
- Cartesian Equation: \( \frac{x - 1}{1} = \frac{y - 2}{-3} = \frac{z - 3}{1} \)