Find the vector equation of the line through (4,3, -1) and parallel to the line :
`vec(r) = (2 hati - hatj + 3 hatk) + lambda (3 hati - hatj + 4 hatk)`.

Options

A. `vec(r) = (4 hati + 3hatj + hatk) + lambda (3 hati - hatj + 4 hatk)`
B. `vec(r) = (4 hati + 3hatj - hatk) + lambda (3 hati - hatj + 4 hatk)`
C. `vec(r) = (4 hati - 3hatj - hatk) + lambda (3 hati - hatj + 4 hatk)`
D. `vec(r) = (4 hati + 3hatj - hatk) + lambda (3 hati - hatj - 4 hatk)`

Answer

To find the vector equation of the line that passes through the point \( (4, 3, -1) \) and is parallel to the line given by the vector equation:

\[ \vec{r} = (2 \hat{i} - \hat{j} + 3 \hat{k}) + \lambda (3 \hat{i} - \hat{j} + 4 \hat{k}), \]

we can follow these steps:

### Step 1: Identify the point through which the line passes
The line we want to find passes through the point \( (4, 3, -1) \). We can represent this point in vector form as:

\[ \vec{A} = 4 \hat{i} + 3 \hat{j} - 1 \hat{k}. \]

### Step 2: Identify the direction vector of the given line
From the given line equation, we can extract the direction vector. The direction vector \( \vec{B} \) is the coefficient of \( \lambda \):

\[ \vec{B} = 3 \hat{i} - \hat{j} + 4 \hat{k}. \]

### Step 3: Write the vector equation of the required line
The vector equation of a line can be written in the form:

\[ \vec{r} = \vec{A} + \lambda \vec{B}, \]

where \( \vec{A} \) is the position vector of a point on the line and \( \vec{B} \) is the direction vector. Substituting the values we have:

\[ \vec{r} = (4 \hat{i} + 3 \hat{j} - 1 \hat{k}) + \lambda (3 \hat{i} - \hat{j} + 4 \hat{k}). \]

### Step 4: Simplify the equation
Now we can express the equation more clearly:

\[ \vec{r} = (4 + 3\lambda) \hat{i} + (3 - \lambda) \hat{j} + (-1 + 4\lambda) \hat{k}. \]

This is the required vector equation of the line.

### Final Answer
Thus, the vector equation of the line through the point \( (4, 3, -1) \) and parallel to the given line is:

\[ \vec{r} = (4 + 3\lambda) \hat{i} + (3 - \lambda) \hat{j} + (-1 + 4\lambda) \hat{k}. \]

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