Find the line of intersection of the planes `vecr.(3hati-hatj+hatk)=1 and vecr.(hati+4hatj-2hatk)=2`

Answer

To find the line of intersection of the planes given by the equations \(\vec{r} \cdot (3\hat{i} - \hat{j} + \hat{k}) = 1\) and \(\vec{r} \cdot (\hat{i} + 4\hat{j} - 2\hat{k}) = 2\), we can follow these steps:

### Step 1: Write the equations of the planes in standard form
The equations of the planes can be rewritten as:
1. \(3x - y + z = 1\) (from the first plane)
2. \(x + 4y - 2z = 2\) (from the second plane)

### Step 2: Set up the system of equations
We now have the following system of equations:
1. \(3x - y + z = 1\) (Equation 1)
2. \(x + 4y - 2z = 2\) (Equation 2)

### Step 3: Solve the system of equations
We can solve these equations simultaneously. Let's express \(z\) in terms of \(x\) and \(y\) from Equation 1:
\[ z = 1 - 3x + y \]

Now, substitute \(z\) into Equation 2:
\[ x + 4y - 2(1 - 3x + y) = 2 \]
Expanding this gives:
\[ x + 4y - 2 + 6x - 2y = 2 \]
Combining like terms:
\[ 7x + 2y - 2 = 2 \]
Thus:
\[ 7x + 2y = 4 \quad \text{(Equation 3)} \]

### Step 4: Express \(y\) in terms of \(x\)
From Equation 3, we can express \(y\):
\[ 2y = 4 - 7x \implies y = 2 - \frac{7}{2}x \]

### Step 5: Substitute \(y\) back to find \(z\)
Now substitute \(y\) back into the expression for \(z\):
\[ z = 1 - 3x + \left(2 - \frac{7}{2}x\right) \]
Simplifying this:
\[ z = 3 - 3x - \frac{7}{2}x = 3 - \left(3 + \frac{7}{2}\right)x = 3 - \frac{13}{2}x \]

### Step 6: Parameterize the line of intersection
Let \(x = t\). Then we have:
\[ y = 2 - \frac{7}{2}t \]
\[ z = 3 - \frac{13}{2}t \]

### Step 7: Write the parametric equations
Thus, the parametric equations of the line of intersection are:
\[ x = t, \quad y = 2 - \frac{7}{2}t, \quad z = 3 - \frac{13}{2}t \]

### Step 8: Write in vector form
The line can be expressed in vector form as:
\[ \vec{r} = \begin{pmatrix} 0 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 1 \\ -\frac{7}{2} \\ -\frac{13}{2} \end{pmatrix} \]

### Final Result
The line of intersection of the two planes is given by:
\[ \vec{r} = \begin{pmatrix} 0 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 1 \\ -\frac{7}{2} \\ -\frac{13}{2} \end{pmatrix} \]