Find the equation of a line passes through the points whose position vectors are `(hati+4hatj+hatk)` and `(2hati-hatj+5hatk)`.

Answer

To find the equation of a line that passes through two points given by their position vectors, we can follow these steps:

### Step 1: Identify the position vectors
Let the two position vectors be:
- \( \mathbf{a} = \hat{i} + 4\hat{j} + \hat{k} \)
- \( \mathbf{b} = 2\hat{i} - \hat{j} + 5\hat{k} \)

### Step 2: Find the direction vector
The direction vector \( \mathbf{d} \) of the line can be found by subtracting the position vector \( \mathbf{a} \) from \( \mathbf{b} \):
\[ \mathbf{d} = \mathbf{b} - \mathbf{a} = (2\hat{i} - \hat{j} + 5\hat{k}) - (\hat{i} + 4\hat{j} + \hat{k}) \]
Calculating this gives:
\[ \mathbf{d} = (2 - 1)\hat{i} + (-1 - 4)\hat{j} + (5 - 1)\hat{k} = \hat{i} - 5\hat{j} + 4\hat{k} \]

### Step 3: Write the equation of the line
The equation of a line in vector form that passes through point \( \mathbf{a} \) and has direction vector \( \mathbf{d} \) is given by:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{d} \]
Substituting the values of \( \mathbf{a} \) and \( \mathbf{d} \):
\[ \mathbf{r} = (\hat{i} + 4\hat{j} + \hat{k}) + \lambda (\hat{i} - 5\hat{j} + 4\hat{k}) \]

### Step 4: Simplify the equation
Now we can simplify this equation:
\[ \mathbf{r} = \hat{i} + 4\hat{j} + \hat{k} + \lambda \hat{i} - 5\lambda \hat{j} + 4\lambda \hat{k} \]
Combining like terms:
\[ \mathbf{r} = (1 + \lambda)\hat{i} + (4 - 5\lambda)\hat{j} + (1 + 4\lambda)\hat{k} \]

### Final Equation
Thus, the equation of the line in vector form is:
\[ \mathbf{r} = (1 + \lambda)\hat{i} + (4 - 5\lambda)\hat{j} + (1 + 4\lambda)\hat{k} \]