Find the equation of straight line parallel to `2hat(i)-hat(j)+3hat(k)` and passing through the point `(5, -2, 4)`.

Answer

To find the equation of a straight line that is parallel to the vector \( \mathbf{v} = 2\hat{i} - \hat{j} + 3\hat{k} \) and passes through the point \( (5, -2, 4) \), we can follow these steps:

### Step 1: Identify the position vector and direction vector
The position vector \( \mathbf{a} \) of the point \( (5, -2, 4) \) is given by:
\[ \mathbf{a} = 5\hat{i} - 2\hat{j} + 4\hat{k} \]
The direction vector \( \mathbf{b} \) is given as:
\[ \mathbf{b} = 2\hat{i} - \hat{j} + 3\hat{k} \]

### Step 2: Write the vector equation of the line
The vector equation of a line passing through a point \( \mathbf{a} \) and parallel to a vector \( \mathbf{b} \) is given by:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \]
Substituting the values of \( \mathbf{a} \) and \( \mathbf{b} \):
\[ \mathbf{r} = (5\hat{i} - 2\hat{j} + 4\hat{k}) + \lambda (2\hat{i} - \hat{j} + 3\hat{k}) \]

### Step 3: Expand the equation
Expanding the equation gives:
\[ \mathbf{r} = 5\hat{i} - 2\hat{j} + 4\hat{k} + \lambda(2\hat{i} - \hat{j} + 3\hat{k}) \]
This can be rewritten as:
\[ \mathbf{r} = (5 + 2\lambda)\hat{i} + (-2 - \lambda)\hat{j} + (4 + 3\lambda)\hat{k} \]

### Step 4: Convert to Cartesian form
Let \( \mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k} \). From the above equation, we can equate the components:
1. \( x = 5 + 2\lambda \)
2. \( y = -2 - \lambda \)
3. \( z = 4 + 3\lambda \)

### Step 5: Solve for \( \lambda \)
From the first equation, we can express \( \lambda \):
\[ \lambda = \frac{x - 5}{2} \]
From the second equation:
\[ \lambda = -2 - y \]
From the third equation:
\[ \lambda = \frac{z - 4}{3} \]

### Step 6: Set the equations equal to each other
Now, we can set these equations equal to each other:
\[ \frac{x - 5}{2} = -2 - y \]
\[ \frac{x - 5}{2} = \frac{z - 4}{3} \]

### Step 7: Write the final equation of the line
We can write the equations collectively as:
\[ \frac{x - 5}{2} = \frac{y + 2}{-1} = \frac{z - 4}{3} \]

### Final Result
Thus, the equation of the straight line in Cartesian form is:
\[ \frac{x - 5}{2} = \frac{y + 2}{-1} = \frac{z - 4}{3} \]