Find the equation of a line passes through the point `hati+hatj+5hatk` and parallel to line joining the points `(2,-4,1)` and `(0,1,3)`.

Answer

To find the equation of a line that passes through the point \( \hat{i} + \hat{j} + 5\hat{k} \) and is parallel to the line joining the points \( (2, -4, 1) \) and \( (0, 1, 3) \), we can follow these steps:

### Step 1: Identify the Points
Let \( A(2, -4, 1) \) and \( B(0, 1, 3) \) be the given points.

### Step 2: Find the Direction Vector
The direction vector \( \vec{AB} \) of the line joining points \( A \) and \( B \) can be calculated as follows:
\[ \vec{AB} = \vec{B} - \vec{A} = (0 - 2)\hat{i} + (1 + 4)\hat{j} + (3 - 1)\hat{k} = -2\hat{i} + 5\hat{j} + 2\hat{k} \]

### Step 3: Write the Equation of the Line
The equation of a line in vector form that passes through a point \( \vec{A} \) and has a direction vector \( \vec{B} \) is given by:
\[ \vec{R} = \vec{A} + k\vec{B} \]
where \( k \) is a scalar parameter.

### Step 4: Substitute the Known Values
Here, \( \vec{A} = \hat{i} + \hat{j} + 5\hat{k} \) and \( \vec{B} = -2\hat{i} + 5\hat{j} + 2\hat{k} \). Substituting these values into the equation gives:
\[ \vec{R} = (\hat{i} + \hat{j} + 5\hat{k}) + k(-2\hat{i} + 5\hat{j} + 2\hat{k}) \]

### Step 5: Simplify the Equation
This can be simplified to:
\[ \vec{R} = \hat{i} + \hat{j} + 5\hat{k} + k(-2\hat{i} + 5\hat{j} + 2\hat{k}) \]
\[ \vec{R} = (\hat{i} - 2k\hat{i}) + (\hat{j} + 5k\hat{j}) + (5\hat{k} + 2k\hat{k}) \]
\[ \vec{R} = (1 - 2k)\hat{i} + (1 + 5k)\hat{j} + (5 + 2k)\hat{k} \]

### Final Equation
Thus, the equation of the line in vector form is:
\[ \vec{R} = (1 - 2k)\hat{i} + (1 + 5k)\hat{j} + (5 + 2k)\hat{k} \]