Find the vector equation of the line which passes through the point (3,4,5) and is parallel the vector `2 hat(i) + 2 hat(j) - 3 hat(k)`.

Options

A. `vec(r) = (3 hati + 4 hatj + 5 hatk ) + lambda (2 hati - 2 hatj - 3 hatk)`
B. `vec(r) = (3 hati + 4 hatj + 5 hatk ) + lambda (2 hati + 2 hatj - 3 hatk)`
C. `vec(r) = (3 hati - 4 hatj + 5 hatk ) + lambda (2 hati + 2 hatj - 3 hatk)`
D. `vec(r) = (3 hati + 4 hatj + 5 hatk ) + lambda (5 hati + 2 hatj - 3 hatk)`

Answer

To find the vector equation of the line that passes through the point (3, 4, 5) and is parallel to the vector \( \vec{b} = 2\hat{i} + 2\hat{j} - 3\hat{k} \), we can follow these steps:

### Step 1: Identify the point and direction vector
The point through which the line passes is given as:
\[ \vec{a} = 3\hat{i} + 4\hat{j} + 5\hat{k} \]
The direction vector of the line is given as:
\[ \vec{b} = 2\hat{i} + 2\hat{j} - 3\hat{k} \]

### Step 2: Write the general equation of a line in vector form
The vector equation of a line can be expressed in the form:
\[ \vec{r} = \vec{a} + \lambda \vec{b} \]
where \( \lambda \) is a scalar parameter.

### Step 3: Substitute the known values into the equation
Substituting \( \vec{a} \) and \( \vec{b} \) into the equation:
\[ \vec{r} = (3\hat{i} + 4\hat{j} + 5\hat{k}) + \lambda (2\hat{i} + 2\hat{j} - 3\hat{k}) \]

### Step 4: Simplify the equation
Distributing \( \lambda \) into the direction vector:
\[ \vec{r} = 3\hat{i} + 4\hat{j} + 5\hat{k} + \lambda(2\hat{i}) + \lambda(2\hat{j}) - \lambda(3\hat{k}) \]
Combining like terms gives:
\[ \vec{r} = (3 + 2\lambda)\hat{i} + (4 + 2\lambda)\hat{j} + (5 - 3\lambda)\hat{k} \]

### Final Answer
Thus, the vector equation of the line is:
\[ \vec{r} = (3 + 2\lambda)\hat{i} + (4 + 2\lambda)\hat{j} + (5 - 3\lambda)\hat{k} \]