Check the co planarity of lines
`vecr=(-3hati+hatj+5hatk)+lambda(-3hati+hatj+5hatk).vecr=(-hati+2hatj+5hatk)+mu(-hati+2hatj+5hatk)`

Answer

To check the coplanarity of the given lines, we will follow these steps:

### Step 1: Identify the lines
The lines are given in vector form:
1. \( \vec{r_1} = (-3\hat{i} + \hat{j} + 5\hat{k}) + \lambda(-3\hat{i} + \hat{j} + 5\hat{k}) \)
2. \( \vec{r_2} = (-\hat{i} + 2\hat{j} + 5\hat{k}) + \mu(-\hat{i} + 2\hat{j} + 5\hat{k}) \)

### Step 2: Write in the form of \( \vec{a_1} + \lambda \vec{b_1} \) and \( \vec{a_2} + \mu \vec{b_2} \)
From the equations, we can identify:
- For the first line:
- \( \vec{a_1} = -3\hat{i} + \hat{j} + 5\hat{k} \)
- \( \vec{b_1} = -3\hat{i} + \hat{j} + 5\hat{k} \)

- For the second line:
- \( \vec{a_2} = -\hat{i} + 2\hat{j} + 5\hat{k} \)
- \( \vec{b_2} = -\hat{i} + 2\hat{j} + 5\hat{k} \)

### Step 3: Calculate \( \vec{a_2} - \vec{a_1} \)
Now, we compute:
\[ \vec{a_2} - \vec{a_1} = (-\hat{i} + 2\hat{j} + 5\hat{k}) - (-3\hat{i} + \hat{j} + 5\hat{k}) \]
Calculating this gives:
\[ \vec{a_2} - \vec{a_1} = (-1 + 3)\hat{i} + (2 - 1)\hat{j} + (5 - 5)\hat{k} = 2\hat{i} + \hat{j} + 0\hat{k} = 2\hat{i} + \hat{j} \]

### Step 4: Calculate the cross product \( \vec{b_1} \times \vec{b_2} \)
Next, we find the cross product of \( \vec{b_1} \) and \( \vec{b_2} \):
\[ \vec{b_1} = -3\hat{i} + \hat{j} + 5\hat{k}, \quad \vec{b_2} = -\hat{i} + 2\hat{j} + 5\hat{k} \]
Using the determinant to find the cross product:
\[ \vec{b_1} \times \vec{b_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -3 & 1 & 5 \\ -1 & 2 & 5 \end{vmatrix} \]
Calculating this determinant:
\[ = \hat{i}(1 \cdot 5 - 5 \cdot 2) - \hat{j}(-3 \cdot 5 - 5 \cdot -1) + \hat{k}(-3 \cdot 2 - 1 \cdot -1) \]
\[ = \hat{i}(5 - 10) - \hat{j}(-15 + 5) + \hat{k}(-6 + 1) \]
\[ = -5\hat{i} + 10\hat{j} - 5\hat{k} \]

### Step 5: Calculate the dot product
Now we compute the dot product:
\[ (\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2}) = (2\hat{i} + \hat{j}) \cdot (-5\hat{i} + 10\hat{j} - 5\hat{k}) \]
Calculating this gives:
\[ = 2 \cdot -5 + 1 \cdot 10 + 0 = -10 + 10 + 0 = 0 \]

### Step 6: Conclusion
Since the dot product is equal to zero, the lines are coplanar.