Find the equation of a line passing through the point (2,0,1) and parallel to the line whose equation is `vecr=(2lambda+3)hati+(7lambda-1)hatj+(-3lambda+2)hatk`

Answer

To find the equation of a line passing through the point (2, 0, 1) and parallel to the given line, we can follow these steps:

### Step 1: Identify the Direction Vector of the Given Line
The equation of the given line is:
\[ \vec{r} = (2\lambda + 3) \hat{i} + (7\lambda - 1) \hat{j} + (-3\lambda + 2) \hat{k} \]
To find the direction vector of this line, we can express it in terms of \(\lambda\):
- The coefficients of \(\lambda\) give us the direction vector of the line:
\[ \vec{d} = \frac{d\vec{r}}{d\lambda} = (2\hat{i} + 7\hat{j} - 3\hat{k}) \]

### Step 2: Write the Position Vector of the Given Point
The point through which our line passes is (2, 0, 1). The position vector of this point is:
\[ \vec{a} = 2\hat{i} + 0\hat{j} + 1\hat{k} = 2\hat{i} + \hat{k} \]

### Step 3: Write the Equation of the Required Line
The equation of a line in vector form can be expressed as:
\[ \vec{r} = \vec{a} + \lambda \vec{d} \]
Substituting \(\vec{a}\) and \(\vec{d}\):
\[ \vec{r} = (2\hat{i} + \hat{k}) + \lambda (2\hat{i} + 7\hat{j} - 3\hat{k}) \]

### Step 4: Simplify the Equation
Now, we can expand and simplify the equation:
\[ \vec{r} = 2\hat{i} + \hat{k} + \lambda (2\hat{i} + 7\hat{j} - 3\hat{k}) \]
\[ = (2 + 2\lambda) \hat{i} + (7\lambda) \hat{j} + (1 - 3\lambda) \hat{k} \]

### Final Equation
Thus, the equation of the line passing through the point (2, 0, 1) and parallel to the given line is:
\[ \vec{r} = (2 + 2\lambda) \hat{i} + (7\lambda) \hat{j} + (1 - 3\lambda) \hat{k} \]

---