Find the equation of the line in vector form passing through the points (1, 1, 0) and (-3, -2, -7).

Answer

To find the equation of the line in vector form passing through the points \( (1, 1, 0) \) and \( (-3, -2, -7) \), we will follow these steps:

### Step 1: Define the position vectors of the points
Let the position vector of the first point \( A \) be:
\[ \vec{A} = 1\hat{i} + 1\hat{j} + 0\hat{k} = \hat{i} + \hat{j} \]

Let the position vector of the second point \( B \) be:
\[ \vec{B} = -3\hat{i} - 2\hat{j} - 7\hat{k} \]

### Step 2: Find the direction vector
The direction vector \( \vec{d} \) of the line can be found by subtracting the position vector \( \vec{A} \) from \( \vec{B} \):
\[ \vec{d} = \vec{B} - \vec{A} = (-3\hat{i} - 2\hat{j} - 7\hat{k}) - (\hat{i} + \hat{j}) \]
Calculating this gives:
\[ \vec{d} = (-3 - 1)\hat{i} + (-2 - 1)\hat{j} + (-7 - 0)\hat{k} = -4\hat{i} - 3\hat{j} - 7\hat{k} \]

### Step 3: Write the equation of the line in vector form
The equation of the line in vector form is given by:
\[ \vec{R} = \vec{A} + \lambda \vec{d} \]
Substituting \( \vec{A} \) and \( \vec{d} \) into this equation:
\[ \vec{R} = (\hat{i} + \hat{j}) + \lambda (-4\hat{i} - 3\hat{j} - 7\hat{k}) \]

### Step 4: Simplify the equation
Expanding this expression:
\[ \vec{R} = \hat{i} + \hat{j} - 4\lambda\hat{i} - 3\lambda\hat{j} - 7\lambda\hat{k} \]
Combining like terms:
\[ \vec{R} = (1 - 4\lambda)\hat{i} + (1 - 3\lambda)\hat{j} - 7\lambda\hat{k} \]

### Final Result
Thus, the equation of the line in vector form is:
\[ \vec{R} = (1 - 4\lambda)\hat{i} + (1 - 3\lambda)\hat{j} - 7\lambda\hat{k} \]